wenzelm@10249: (*  Title:      HOL/Library/Multiset.thy
paulson@15072:     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
blanchet@55129:     Author:     Andrei Popescu, TU Muenchen
blanchet@59813:     Author:     Jasmin Blanchette, Inria, LORIA, MPII
blanchet@59813:     Author:     Dmitriy Traytel, TU Muenchen
blanchet@59813:     Author:     Mathias Fleury, MPII
wenzelm@10249: *)
wenzelm@10249: 
wenzelm@60500: section \<open>(Finite) multisets\<close>
wenzelm@10249: 
nipkow@15131: theory Multiset
haftmann@51599: imports Main
nipkow@15131: begin
wenzelm@10249: 
wenzelm@60500: subsection \<open>The type of multisets\<close>
wenzelm@10249: 
wenzelm@60606: definition "multiset = {f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}"
wenzelm@60606: 
wenzelm@60606: typedef 'a multiset = "multiset :: ('a \<Rightarrow> nat) set"
haftmann@34943:   morphisms count Abs_multiset
wenzelm@45694:   unfolding multiset_def
wenzelm@10249: proof
wenzelm@45694:   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
wenzelm@10249: qed
wenzelm@10249: 
bulwahn@47429: setup_lifting type_definition_multiset
wenzelm@19086: 
wenzelm@60606: abbreviation Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool"  ("(_/ :# _)" [50, 51] 50) where
kleing@25610:   "a :# M == 0 < count M a"
kleing@25610: 
wenzelm@26145: notation (xsymbols)
wenzelm@26145:   Melem (infix "\<in>#" 50)
wenzelm@10249: 
wenzelm@60606: lemma multiset_eq_iff: "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
nipkow@39302:   by (simp only: count_inject [symmetric] fun_eq_iff)
haftmann@34943: 
wenzelm@60606: lemma multiset_eqI: "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
nipkow@39302:   using multiset_eq_iff by auto
haftmann@34943: 
wenzelm@60606: text \<open>Preservation of the representing set @{term multiset}.\<close>
wenzelm@60606: 
wenzelm@60606: lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
haftmann@34943:   by (simp add: multiset_def)
haftmann@34943: 
wenzelm@60606: lemma only1_in_multiset: "(\<lambda>b. if b = a then n else 0) \<in> multiset"
haftmann@34943:   by (simp add: multiset_def)
haftmann@34943: 
wenzelm@60606: lemma union_preserves_multiset: "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
haftmann@34943:   by (simp add: multiset_def)
haftmann@34943: 
haftmann@34943: lemma diff_preserves_multiset:
haftmann@34943:   assumes "M \<in> multiset"
haftmann@34943:   shows "(\<lambda>a. M a - N a) \<in> multiset"
haftmann@34943: proof -
haftmann@34943:   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
haftmann@34943:     by auto
haftmann@34943:   with assms show ?thesis
haftmann@34943:     by (auto simp add: multiset_def intro: finite_subset)
haftmann@34943: qed
haftmann@34943: 
haftmann@41069: lemma filter_preserves_multiset:
haftmann@34943:   assumes "M \<in> multiset"
haftmann@34943:   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
haftmann@34943: proof -
haftmann@34943:   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
haftmann@34943:     by auto
haftmann@34943:   with assms show ?thesis
haftmann@34943:     by (auto simp add: multiset_def intro: finite_subset)
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemmas in_multiset = const0_in_multiset only1_in_multiset
haftmann@41069:   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
haftmann@34943: 
haftmann@34943: 
wenzelm@60500: subsection \<open>Representing multisets\<close>
wenzelm@60500: 
wenzelm@60500: text \<open>Multiset enumeration\<close>
haftmann@34943: 
huffman@48008: instantiation multiset :: (type) cancel_comm_monoid_add
haftmann@25571: begin
haftmann@25571: 
bulwahn@47429: lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
bulwahn@47429: by (rule const0_in_multiset)
haftmann@25571: 
haftmann@34943: abbreviation Mempty :: "'a multiset" ("{#}") where
haftmann@34943:   "Mempty \<equiv> 0"
haftmann@25571: 
wenzelm@60606: lift_definition plus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
bulwahn@47429: by (rule union_preserves_multiset)
haftmann@25571: 
wenzelm@60606: lift_definition minus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
haftmann@59815: by (rule diff_preserves_multiset)
haftmann@59815: 
huffman@48008: instance
wenzelm@60678:   by (standard; transfer; simp add: fun_eq_iff)
haftmann@25571: 
haftmann@25571: end
wenzelm@10249: 
wenzelm@60606: lift_definition single :: "'a \<Rightarrow> 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
bulwahn@47429: by (rule only1_in_multiset)
kleing@15869: 
wenzelm@26145: syntax
wenzelm@60606:   "_multiset" :: "args \<Rightarrow> 'a multiset"    ("{#(_)#}")
nipkow@25507: translations
nipkow@25507:   "{#x, xs#}" == "{#x#} + {#xs#}"
nipkow@25507:   "{#x#}" == "CONST single x"
nipkow@25507: 
haftmann@34943: lemma count_empty [simp]: "count {#} a = 0"
bulwahn@47429:   by (simp add: zero_multiset.rep_eq)
wenzelm@10249: 
haftmann@34943: lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
bulwahn@47429:   by (simp add: single.rep_eq)
nipkow@29901: 
wenzelm@10249: 
wenzelm@60500: subsection \<open>Basic operations\<close>
wenzelm@60500: 
wenzelm@60500: subsubsection \<open>Union\<close>
wenzelm@10249: 
haftmann@34943: lemma count_union [simp]: "count (M + N) a = count M a + count N a"
bulwahn@47429:   by (simp add: plus_multiset.rep_eq)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@60500: subsubsection \<open>Difference\<close>
wenzelm@10249: 
haftmann@49388: instantiation multiset :: (type) comm_monoid_diff
haftmann@34943: begin
haftmann@34943: 
haftmann@49388: instance
wenzelm@60678:   by (standard; transfer; simp add: fun_eq_iff)
haftmann@34943: 
haftmann@34943: end
haftmann@34943: 
haftmann@34943: lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
bulwahn@47429:   by (simp add: minus_multiset.rep_eq)
haftmann@34943: 
wenzelm@17161: lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
haftmann@52289:   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
nipkow@36903: 
nipkow@36903: lemma diff_cancel[simp]: "A - A = {#}"
haftmann@52289:   by (fact Groups.diff_cancel)
wenzelm@10249: 
nipkow@36903: lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
haftmann@52289:   by (fact add_diff_cancel_right')
wenzelm@10249: 
nipkow@36903: lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
haftmann@52289:   by (fact add_diff_cancel_left')
haftmann@34943: 
haftmann@52289: lemma diff_right_commute:
wenzelm@60606:   fixes M N Q :: "'a multiset"
wenzelm@60606:   shows "M - N - Q = M - Q - N"
haftmann@52289:   by (fact diff_right_commute)
haftmann@52289: 
haftmann@52289: lemma diff_add:
wenzelm@60606:   fixes M N Q :: "'a multiset"
wenzelm@60606:   shows "M - (N + Q) = M - N - Q"
haftmann@52289:   by (rule sym) (fact diff_diff_add)
blanchet@58425: 
wenzelm@60606: lemma insert_DiffM: "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
nipkow@39302:   by (clarsimp simp: multiset_eq_iff)
haftmann@34943: 
wenzelm@60606: lemma insert_DiffM2 [simp]: "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
nipkow@39302:   by (clarsimp simp: multiset_eq_iff)
haftmann@34943: 
wenzelm@60606: lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
wenzelm@60606: lemma diff_union_single_conv: "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
nipkow@39302:   by (simp add: multiset_eq_iff)
bulwahn@26143: 
wenzelm@10249: 
wenzelm@60500: subsubsection \<open>Equality of multisets\<close>
haftmann@34943: 
haftmann@34943: lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
nipkow@39302:   by (simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
haftmann@34943: lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
wenzelm@60606: lemma diff_single_trivial: "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
nipkow@39302:   by (auto simp add: multiset_eq_iff)
haftmann@34943: 
wenzelm@60606: lemma diff_single_eq_union: "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
haftmann@34943:   by auto
haftmann@34943: 
wenzelm@60606: lemma union_single_eq_diff: "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
haftmann@34943:   by (auto dest: sym)
haftmann@34943: 
wenzelm@60606: lemma union_single_eq_member: "M + {#x#} = N \<Longrightarrow> x \<in># N"
haftmann@34943:   by auto
haftmann@34943: 
wenzelm@60606: lemma union_is_single: "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}"
wenzelm@60606:   (is "?lhs = ?rhs")
wenzelm@46730: proof
wenzelm@60606:   show ?lhs if ?rhs using that by auto
wenzelm@60606:   show ?rhs if ?lhs
wenzelm@60606:     using that by (simp add: multiset_eq_iff split: if_splits) (metis add_is_1)
haftmann@34943: qed
haftmann@34943: 
wenzelm@60606: lemma single_is_union: "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
haftmann@34943:   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
haftmann@34943: 
haftmann@34943: lemma add_eq_conv_diff:
wenzelm@60606:   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"
wenzelm@60606:   (is "?lhs \<longleftrightarrow> ?rhs")
nipkow@44890: (* shorter: by (simp add: multiset_eq_iff) fastforce *)
haftmann@34943: proof
wenzelm@60606:   show ?lhs if ?rhs
wenzelm@60606:     using that
wenzelm@60606:     by (auto simp add: add.assoc add.commute [of "{#b#}"])
wenzelm@60606:       (drule sym, simp add: add.assoc [symmetric])
wenzelm@60606:   show ?rhs if ?lhs
haftmann@34943:   proof (cases "a = b")
wenzelm@60500:     case True with \<open>?lhs\<close> show ?thesis by simp
haftmann@34943:   next
haftmann@34943:     case False
wenzelm@60500:     from \<open>?lhs\<close> have "a \<in># N + {#b#}" by (rule union_single_eq_member)
haftmann@34943:     with False have "a \<in># N" by auto
wenzelm@60500:     moreover from \<open>?lhs\<close> have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
haftmann@34943:     moreover note False
haftmann@34943:     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
haftmann@34943:   qed
haftmann@34943: qed
haftmann@34943: 
blanchet@58425: lemma insert_noteq_member:
haftmann@34943:   assumes BC: "B + {#b#} = C + {#c#}"
haftmann@34943:    and bnotc: "b \<noteq> c"
haftmann@34943:   shows "c \<in># B"
haftmann@34943: proof -
haftmann@34943:   have "c \<in># C + {#c#}" by simp
haftmann@34943:   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
haftmann@34943:   then have "c \<in># B + {#b#}" using BC by simp
haftmann@34943:   then show "c \<in># B" using nc by simp
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: lemma add_eq_conv_ex:
haftmann@34943:   "(M + {#a#} = N + {#b#}) =
haftmann@34943:     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
haftmann@34943:   by (auto simp add: add_eq_conv_diff)
haftmann@34943: 
wenzelm@60606: lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
wenzelm@60678:   by (rule exI [where x = "M - {#x#}"]) simp
haftmann@51600: 
blanchet@58425: lemma multiset_add_sub_el_shuffle:
wenzelm@60606:   assumes "c \<in># B"
wenzelm@60606:     and "b \<noteq> c"
haftmann@58098:   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
haftmann@58098: proof -
wenzelm@60500:   from \<open>c \<in># B\<close> obtain A where B: "B = A + {#c#}"
haftmann@58098:     by (blast dest: multi_member_split)
haftmann@58098:   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
blanchet@58425:   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
haftmann@58098:     by (simp add: ac_simps)
haftmann@58098:   then show ?thesis using B by simp
haftmann@58098: qed
haftmann@58098: 
haftmann@34943: 
wenzelm@60500: subsubsection \<open>Pointwise ordering induced by count\<close>
haftmann@34943: 
Mathias@60397: definition subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
Mathias@60400: "subseteq_mset A B = (\<forall>a. count A a \<le> count B a)"
Mathias@60397: 
Mathias@60397: definition subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
Mathias@60400: "subset_mset A B = (A <=# B \<and> A \<noteq> B)"
Mathias@60397: 
Mathias@60397: notation subseteq_mset (infix "\<le>#" 50)
Mathias@60397: notation (xsymbols) subseteq_mset (infix "\<subseteq>#" 50)
Mathias@60397: 
Mathias@60397: notation (xsymbols) subset_mset (infix "\<subset>#" 50)
Mathias@60397: 
wenzelm@60606: interpretation subset_mset: ordered_ab_semigroup_add_imp_le "op +" "op -" "op \<subseteq>#" "op \<subset>#"
wenzelm@60678:   by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)
blanchet@59986: 
wenzelm@60606: lemma mset_less_eqI: "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le># B"
Mathias@60397:   by (simp add: subseteq_mset_def)
haftmann@34943: 
wenzelm@60606: lemma mset_le_exists_conv: "(A::'a multiset) \<le># B \<longleftrightarrow> (\<exists>C. B = A + C)"
wenzelm@60678:   unfolding subseteq_mset_def
wenzelm@60678:   apply (rule iffI)
wenzelm@60678:    apply (rule exI [where x = "B - A"])
wenzelm@60678:    apply (auto intro: multiset_eq_iff [THEN iffD2])
wenzelm@60678:   done
haftmann@34943: 
Mathias@60397: interpretation subset_mset: ordered_cancel_comm_monoid_diff  "op +" "op -" 0 "op \<le>#" "op <#"
wenzelm@60678:   by standard (simp, fact mset_le_exists_conv)
haftmann@52289: 
wenzelm@60606: lemma mset_le_mono_add_right_cancel [simp]: "(A::'a multiset) + C \<le># B + C \<longleftrightarrow> A \<le># B"
Mathias@60397:   by (fact subset_mset.add_le_cancel_right)
haftmann@34943: 
wenzelm@60606: lemma mset_le_mono_add_left_cancel [simp]: "C + (A::'a multiset) \<le># C + B \<longleftrightarrow> A \<le># B"
Mathias@60397:   by (fact subset_mset.add_le_cancel_left)
haftmann@35268: 
wenzelm@60606: lemma mset_le_mono_add: "(A::'a multiset) \<le># B \<Longrightarrow> C \<le># D \<Longrightarrow> A + C \<le># B + D"
Mathias@60397:   by (fact subset_mset.add_mono)
haftmann@34943: 
wenzelm@60606: lemma mset_le_add_left [simp]: "(A::'a multiset) \<le># A + B"
Mathias@60397:   unfolding subseteq_mset_def by auto
haftmann@35268: 
wenzelm@60606: lemma mset_le_add_right [simp]: "B \<le># (A::'a multiset) + B"
Mathias@60397:   unfolding subseteq_mset_def by auto
haftmann@34943: 
wenzelm@60607: lemma mset_le_single: "a \<in># B \<Longrightarrow> {#a#} \<le># B"
Mathias@60397:   by (simp add: subseteq_mset_def)
haftmann@34943: 
haftmann@35268: lemma multiset_diff_union_assoc:
wenzelm@60606:   fixes A B C D :: "'a multiset"
wenzelm@60606:   shows "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
Mathias@60397:   by (simp add: subset_mset.diff_add_assoc)
haftmann@34943: 
haftmann@34943: lemma mset_le_multiset_union_diff_commute:
wenzelm@60606:   fixes A B C D :: "'a multiset"
wenzelm@60606:   shows "B \<le># A \<Longrightarrow> A - B + C = A + C - B"
Mathias@60397: by (simp add: subset_mset.add_diff_assoc2)
Mathias@60397: 
Mathias@60397: lemma diff_le_self[simp]: "(M::'a multiset) - N \<le># M"
Mathias@60397: by(simp add: subseteq_mset_def)
Mathias@60397: 
Mathias@60397: lemma mset_lessD: "A <# B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
Mathias@60397: apply (clarsimp simp: subset_mset_def subseteq_mset_def)
wenzelm@60678: apply (erule allE [where x = x])
haftmann@34943: apply auto
haftmann@34943: done
haftmann@34943: 
Mathias@60397: lemma mset_leD: "A \<le># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
Mathias@60397: apply (clarsimp simp: subset_mset_def subseteq_mset_def)
wenzelm@60678: apply (erule allE [where x = x])
haftmann@34943: apply auto
haftmann@34943: done
blanchet@58425: 
Mathias@60397: lemma mset_less_insertD: "(A + {#x#} <# B) \<Longrightarrow> (x \<in># B \<and> A <# B)"
haftmann@34943: apply (rule conjI)
haftmann@34943:  apply (simp add: mset_lessD)
Mathias@60397: apply (clarsimp simp: subset_mset_def subseteq_mset_def)
haftmann@34943: apply safe
haftmann@34943:  apply (erule_tac x = a in allE)
haftmann@34943:  apply (auto split: split_if_asm)
haftmann@34943: done
haftmann@34943: 
Mathias@60397: lemma mset_le_insertD: "(A + {#x#} \<le># B) \<Longrightarrow> (x \<in># B \<and> A \<le># B)"
haftmann@34943: apply (rule conjI)
haftmann@34943:  apply (simp add: mset_leD)
Mathias@60397: apply (force simp: subset_mset_def subseteq_mset_def split: split_if_asm)
haftmann@34943: done
haftmann@34943: 
Mathias@60397: lemma mset_less_of_empty[simp]: "A <# {#} \<longleftrightarrow> False"
Mathias@60397:   by (auto simp add: subseteq_mset_def subset_mset_def multiset_eq_iff)
Mathias@60397: 
Mathias@60397: lemma empty_le[simp]: "{#} \<le># A"
nipkow@55808:   unfolding mset_le_exists_conv by auto
nipkow@55808: 
Mathias@60397: lemma le_empty[simp]: "(M \<le># {#}) = (M = {#})"
nipkow@55808:   unfolding mset_le_exists_conv by auto
nipkow@55808: 
Mathias@60397: lemma multi_psub_of_add_self[simp]: "A <# A + {#x#}"
Mathias@60397:   by (auto simp: subset_mset_def subseteq_mset_def)
Mathias@60397: 
Mathias@60397: lemma multi_psub_self[simp]: "(A::'a multiset) <# A = False"
haftmann@35268:   by simp
haftmann@34943: 
Mathias@60397: lemma mset_less_add_bothsides: "N + {#x#} <# M + {#x#} \<Longrightarrow> N <# M"
Mathias@60397:   by (fact subset_mset.add_less_imp_less_right)
haftmann@35268: 
wenzelm@60606: lemma mset_less_empty_nonempty: "{#} <# S \<longleftrightarrow> S \<noteq> {#}"
Mathias@60397:   by (auto simp: subset_mset_def subseteq_mset_def)
haftmann@35268: 
wenzelm@60606: lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} <# B"
Mathias@60397:   by (auto simp: subset_mset_def subseteq_mset_def multiset_eq_iff)
haftmann@35268: 
haftmann@35268: 
wenzelm@60500: subsubsection \<open>Intersection\<close>
haftmann@35268: 
Mathias@60397: definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
Mathias@60397:   multiset_inter_def: "inf_subset_mset A B = A - (A - B)"
Mathias@60397: 
Mathias@60397: interpretation subset_mset: semilattice_inf inf_subset_mset "op \<le>#" "op <#"
wenzelm@46921: proof -
wenzelm@60678:   have [simp]: "m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" for m n q :: nat
wenzelm@60678:     by arith
wenzelm@60678:   show "class.semilattice_inf op #\<inter> op \<le># op <#"
wenzelm@60678:     by standard (auto simp add: multiset_inter_def subseteq_mset_def)
haftmann@35268: qed
haftmann@35268: 
haftmann@34943: 
haftmann@41069: lemma multiset_inter_count [simp]:
wenzelm@60606:   fixes A B :: "'a multiset"
wenzelm@60606:   shows "count (A #\<inter> B) x = min (count A x) (count B x)"
bulwahn@47429:   by (simp add: multiset_inter_def)
haftmann@35268: 
haftmann@35268: lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
wenzelm@46730:   by (rule multiset_eqI) auto
haftmann@34943: 
haftmann@35268: lemma multiset_union_diff_commute:
haftmann@35268:   assumes "B #\<inter> C = {#}"
haftmann@35268:   shows "A + B - C = A - C + B"
nipkow@39302: proof (rule multiset_eqI)
haftmann@35268:   fix x
haftmann@35268:   from assms have "min (count B x) (count C x) = 0"
wenzelm@46730:     by (auto simp add: multiset_eq_iff)
haftmann@35268:   then have "count B x = 0 \<or> count C x = 0"
haftmann@35268:     by auto
haftmann@35268:   then show "count (A + B - C) x = count (A - C + B) x"
haftmann@35268:     by auto
haftmann@35268: qed
haftmann@35268: 
wenzelm@60606: lemma empty_inter [simp]: "{#} #\<inter> M = {#}"
haftmann@51600:   by (simp add: multiset_eq_iff)
haftmann@51600: 
wenzelm@60606: lemma inter_empty [simp]: "M #\<inter> {#} = {#}"
haftmann@51600:   by (simp add: multiset_eq_iff)
haftmann@51600: 
wenzelm@60606: lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
haftmann@51600:   by (simp add: multiset_eq_iff)
haftmann@51600: 
wenzelm@60606: lemma inter_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
haftmann@51600:   by (simp add: multiset_eq_iff)
haftmann@51600: 
wenzelm@60606: lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
haftmann@51600:   by (simp add: multiset_eq_iff)
haftmann@51600: 
wenzelm@60606: lemma inter_add_right2: "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
haftmann@51600:   by (simp add: multiset_eq_iff)
haftmann@51600: 
haftmann@35268: 
wenzelm@60500: subsubsection \<open>Bounded union\<close>
wenzelm@60678: 
wenzelm@60678: definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "#\<union>" 70)
wenzelm@60678:   where "sup_subset_mset A B = A + (B - A)"
Mathias@60397: 
Mathias@60397: interpretation subset_mset: semilattice_sup sup_subset_mset "op \<le>#" "op <#"
haftmann@51623: proof -
wenzelm@60678:   have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat
wenzelm@60678:     by arith
Mathias@60397:   show "class.semilattice_sup op #\<union> op \<le># op <#"
wenzelm@60678:     by standard (auto simp add: sup_subset_mset_def subseteq_mset_def)
haftmann@51623: qed
haftmann@51623: 
wenzelm@60606: lemma sup_subset_mset_count [simp]: "count (A #\<union> B) x = max (count A x) (count B x)"
Mathias@60397:   by (simp add: sup_subset_mset_def)
haftmann@51623: 
wenzelm@60606: lemma empty_sup [simp]: "{#} #\<union> M = M"
haftmann@51623:   by (simp add: multiset_eq_iff)
haftmann@51623: 
wenzelm@60606: lemma sup_empty [simp]: "M #\<union> {#} = M"
haftmann@51623:   by (simp add: multiset_eq_iff)
haftmann@51623: 
wenzelm@60606: lemma sup_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
haftmann@51623:   by (simp add: multiset_eq_iff)
haftmann@51623: 
wenzelm@60606: lemma sup_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
haftmann@51623:   by (simp add: multiset_eq_iff)
haftmann@51623: 
wenzelm@60606: lemma sup_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
haftmann@51623:   by (simp add: multiset_eq_iff)
haftmann@51623: 
wenzelm@60606: lemma sup_add_right2: "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
haftmann@51623:   by (simp add: multiset_eq_iff)
haftmann@51623: 
wenzelm@60500: subsubsection \<open>Subset is an order\<close>
Mathias@60397: interpretation subset_mset: order "op \<le>#" "op <#" by unfold_locales auto
haftmann@51623: 
wenzelm@60500: subsubsection \<open>Filter (with comprehension syntax)\<close>
wenzelm@60500: 
wenzelm@60500: text \<open>Multiset comprehension\<close>
haftmann@41069: 
nipkow@59998: lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"
nipkow@59998: is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
bulwahn@47429: by (rule filter_preserves_multiset)
haftmann@35268: 
wenzelm@60606: lemma count_filter_mset [simp]: "count (filter_mset P M) a = (if P a then count M a else 0)"
nipkow@59998:   by (simp add: filter_mset.rep_eq)
nipkow@59998: 
wenzelm@60606: lemma filter_empty_mset [simp]: "filter_mset P {#} = {#}"
nipkow@59998:   by (rule multiset_eqI) simp
nipkow@59998: 
wenzelm@60606: lemma filter_single_mset [simp]: "filter_mset P {#x#} = (if P x then {#x#} else {#})"
nipkow@39302:   by (rule multiset_eqI) simp
haftmann@35268: 
wenzelm@60606: lemma filter_union_mset [simp]: "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
haftmann@41069:   by (rule multiset_eqI) simp
haftmann@41069: 
wenzelm@60606: lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
nipkow@39302:   by (rule multiset_eqI) simp
haftmann@35268: 
wenzelm@60606: lemma filter_inter_mset [simp]: "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N"
haftmann@41069:   by (rule multiset_eqI) simp
haftmann@41069: 
Mathias@60397: lemma multiset_filter_subset[simp]: "filter_mset f M \<le># M"
Mathias@60397:   by (simp add: mset_less_eqI)
Mathias@60397: 
wenzelm@60606: lemma multiset_filter_mono:
wenzelm@60606:   assumes "A \<le># B"
Mathias@60397:   shows "filter_mset f A \<le># filter_mset f B"
blanchet@58035: proof -
blanchet@58035:   from assms[unfolded mset_le_exists_conv]
blanchet@58035:   obtain C where B: "B = A + C" by auto
blanchet@58035:   show ?thesis unfolding B by auto
blanchet@58035: qed
blanchet@58035: 
haftmann@41069: syntax
haftmann@41069:   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
haftmann@41069: syntax (xsymbol)
haftmann@41069:   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
haftmann@41069: translations
nipkow@59998:   "{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M"
haftmann@41069: 
wenzelm@10249: 
wenzelm@60500: subsubsection \<open>Set of elements\<close>
wenzelm@10249: 
wenzelm@60606: definition set_mset :: "'a multiset \<Rightarrow> 'a set"
wenzelm@60607:   where "set_mset M = {x. x \<in># M}"
nipkow@60495: 
nipkow@60495: lemma set_mset_empty [simp]: "set_mset {#} = {}"
nipkow@60495: by (simp add: set_mset_def)
nipkow@60495: 
nipkow@60495: lemma set_mset_single [simp]: "set_mset {#b#} = {b}"
nipkow@60495: by (simp add: set_mset_def)
nipkow@60495: 
nipkow@60495: lemma set_mset_union [simp]: "set_mset (M + N) = set_mset M \<union> set_mset N"
nipkow@60495: by (auto simp add: set_mset_def)
nipkow@60495: 
nipkow@60495: lemma set_mset_eq_empty_iff [simp]: "(set_mset M = {}) = (M = {#})"
nipkow@60495: by (auto simp add: set_mset_def multiset_eq_iff)
nipkow@60495: 
wenzelm@60607: lemma mem_set_mset_iff [simp]: "(x \<in> set_mset M) = (x \<in># M)"
nipkow@60495: by (auto simp add: set_mset_def)
nipkow@60495: 
wenzelm@60607: lemma set_mset_filter [simp]: "set_mset {# x\<in>#M. P x #} = set_mset M \<inter> {x. P x}"
nipkow@60495: by (auto simp add: set_mset_def)
nipkow@60495: 
nipkow@60495: lemma finite_set_mset [iff]: "finite (set_mset M)"
nipkow@60495:   using count [of M] by (simp add: multiset_def set_mset_def)
haftmann@34943: 
wenzelm@60607: lemma finite_Collect_mem [iff]: "finite {x. x \<in># M}"
nipkow@60495:   unfolding set_mset_def[symmetric] by simp
nipkow@60495: 
nipkow@60495: lemma set_mset_mono: "A \<le># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
nipkow@60495:   by (metis mset_leD subsetI mem_set_mset_iff)
nipkow@60495: 
nipkow@60495: lemma ball_set_mset_iff: "(\<forall>x \<in> set_mset M. P x) \<longleftrightarrow> (\<forall>x. x \<in># M \<longrightarrow> P x)"
blanchet@59813:   by auto
blanchet@59813: 
blanchet@59813: 
wenzelm@60500: subsubsection \<open>Size\<close>
wenzelm@10249: 
blanchet@56656: definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
blanchet@56656: 
blanchet@56656: lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
blanchet@56656:   by (auto simp: wcount_def add_mult_distrib)
blanchet@56656: 
blanchet@56656: definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
nipkow@60495:   "size_multiset f M = setsum (wcount f M) (set_mset M)"
blanchet@56656: 
blanchet@56656: lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
blanchet@56656: 
wenzelm@60606: instantiation multiset :: (type) size
wenzelm@60606: begin
wenzelm@60606: 
blanchet@56656: definition size_multiset where
blanchet@56656:   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
haftmann@34943: instance ..
wenzelm@60606: 
haftmann@34943: end
haftmann@34943: 
blanchet@56656: lemmas size_multiset_overloaded_eq =
blanchet@56656:   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
blanchet@56656: 
blanchet@56656: lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
blanchet@56656: by (simp add: size_multiset_def)
blanchet@56656: 
haftmann@28708: lemma size_empty [simp]: "size {#} = 0"
blanchet@56656: by (simp add: size_multiset_overloaded_def)
blanchet@56656: 
blanchet@56656: lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
blanchet@56656: by (simp add: size_multiset_eq)
wenzelm@10249: 
haftmann@28708: lemma size_single [simp]: "size {#b#} = 1"
blanchet@56656: by (simp add: size_multiset_overloaded_def)
blanchet@56656: 
blanchet@56656: lemma setsum_wcount_Int:
nipkow@60495:   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_mset N) = setsum (wcount f N) A"
nipkow@26178: apply (induct rule: finite_induct)
nipkow@26178:  apply simp
nipkow@60495: apply (simp add: Int_insert_left set_mset_def wcount_def)
blanchet@56656: done
blanchet@56656: 
blanchet@56656: lemma size_multiset_union [simp]:
blanchet@56656:   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
haftmann@57418: apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
blanchet@56656: apply (subst Int_commute)
blanchet@56656: apply (simp add: setsum_wcount_Int)
nipkow@26178: done
wenzelm@10249: 
haftmann@28708: lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
blanchet@56656: by (auto simp add: size_multiset_overloaded_def)
blanchet@56656: 
blanchet@56656: lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
blanchet@56656: by (auto simp add: size_multiset_eq multiset_eq_iff)
wenzelm@10249: 
wenzelm@17161: lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
blanchet@56656: by (auto simp add: size_multiset_overloaded_def)
nipkow@26016: 
nipkow@26016: lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
nipkow@26178: by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
wenzelm@10249: 
wenzelm@60607: lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M"
blanchet@56656: apply (unfold size_multiset_overloaded_eq)
nipkow@26178: apply (drule setsum_SucD)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
haftmann@34943: lemma size_eq_Suc_imp_eq_union:
haftmann@34943:   assumes "size M = Suc n"
haftmann@34943:   shows "\<exists>a N. M = N + {#a#}"
haftmann@34943: proof -
haftmann@34943:   from assms obtain a where "a \<in># M"
haftmann@34943:     by (erule size_eq_Suc_imp_elem [THEN exE])
haftmann@34943:   then have "M = M - {#a#} + {#a#}" by simp
haftmann@34943:   then show ?thesis by blast
nipkow@23611: qed
kleing@15869: 
wenzelm@60606: lemma size_mset_mono:
wenzelm@60606:   fixes A B :: "'a multiset"
wenzelm@60606:   assumes "A \<le># B"
wenzelm@60606:   shows "size A \<le> size B"
nipkow@59949: proof -
nipkow@59949:   from assms[unfolded mset_le_exists_conv]
nipkow@59949:   obtain C where B: "B = A + C" by auto
wenzelm@60606:   show ?thesis unfolding B by (induct C) auto
nipkow@59949: qed
nipkow@59949: 
nipkow@59998: lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M"
nipkow@59949: by (rule size_mset_mono[OF multiset_filter_subset])
nipkow@59949: 
nipkow@59949: lemma size_Diff_submset:
Mathias@60397:   "M \<le># M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
nipkow@59949: by (metis add_diff_cancel_left' size_union mset_le_exists_conv)
nipkow@26016: 
wenzelm@60500: subsection \<open>Induction and case splits\<close>
wenzelm@10249: 
wenzelm@18258: theorem multiset_induct [case_names empty add, induct type: multiset]:
huffman@48009:   assumes empty: "P {#}"
huffman@48009:   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
huffman@48009:   shows "P M"
huffman@48009: proof (induct n \<equiv> "size M" arbitrary: M)
huffman@48009:   case 0 thus "P M" by (simp add: empty)
huffman@48009: next
huffman@48009:   case (Suc k)
huffman@48009:   obtain N x where "M = N + {#x#}"
wenzelm@60500:     using \<open>Suc k = size M\<close> [symmetric]
huffman@48009:     using size_eq_Suc_imp_eq_union by fast
huffman@48009:   with Suc add show "P M" by simp
wenzelm@10249: qed
wenzelm@10249: 
kleing@25610: lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
nipkow@26178: by (induct M) auto
kleing@25610: 
wenzelm@55913: lemma multiset_cases [cases type]:
wenzelm@55913:   obtains (empty) "M = {#}"
wenzelm@55913:     | (add) N x where "M = N + {#x#}"
wenzelm@55913:   using assms by (induct M) simp_all
kleing@25610: 
haftmann@34943: lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
haftmann@34943: by (cases "B = {#}") (auto dest: multi_member_split)
haftmann@34943: 
wenzelm@60607: lemma multiset_partition: "M = {# x\<in>#M. P x #} + {# x\<in>#M. \<not> P x #}"
nipkow@39302: apply (subst multiset_eq_iff)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
Mathias@60397: lemma mset_less_size: "(A::'a multiset) <# B \<Longrightarrow> size A < size B"
haftmann@34943: proof (induct A arbitrary: B)
haftmann@34943:   case (empty M)
haftmann@34943:   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
blanchet@58425:   then obtain M' x where "M = M' + {#x#}"
haftmann@34943:     by (blast dest: multi_nonempty_split)
haftmann@34943:   then show ?case by simp
haftmann@34943: next
haftmann@34943:   case (add S x T)
Mathias@60397:   have IH: "\<And>B. S <# B \<Longrightarrow> size S < size B" by fact
Mathias@60397:   have SxsubT: "S + {#x#} <# T" by fact
Mathias@60397:   then have "x \<in># T" and "S <# T" by (auto dest: mset_less_insertD)
blanchet@58425:   then obtain T' where T: "T = T' + {#x#}"
haftmann@34943:     by (blast dest: multi_member_split)
Mathias@60397:   then have "S <# T'" using SxsubT
haftmann@34943:     by (blast intro: mset_less_add_bothsides)
haftmann@34943:   then have "size S < size T'" using IH by simp
haftmann@34943:   then show ?case using T by simp
haftmann@34943: qed
haftmann@34943: 
haftmann@34943: 
nipkow@59949: lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
nipkow@59949: by (cases M) auto
nipkow@59949: 
wenzelm@60500: subsubsection \<open>Strong induction and subset induction for multisets\<close>
wenzelm@60500: 
wenzelm@60500: text \<open>Well-foundedness of strict subset relation\<close>
haftmann@58098: 
Mathias@60397: lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M <# N}"
haftmann@34943: apply (rule wf_measure [THEN wf_subset, where f1=size])
haftmann@34943: apply (clarsimp simp: measure_def inv_image_def mset_less_size)
haftmann@34943: done
haftmann@34943: 
haftmann@34943: lemma full_multiset_induct [case_names less]:
Mathias@60397: assumes ih: "\<And>B. \<forall>(A::'a multiset). A <# B \<longrightarrow> P A \<Longrightarrow> P B"
haftmann@34943: shows "P B"
haftmann@58098: apply (rule wf_less_mset_rel [THEN wf_induct])
haftmann@58098: apply (rule ih, auto)
haftmann@34943: done
haftmann@34943: 
haftmann@34943: lemma multi_subset_induct [consumes 2, case_names empty add]:
wenzelm@60606:   assumes "F \<le># A"
wenzelm@60606:     and empty: "P {#}"
wenzelm@60606:     and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
wenzelm@60606:   shows "P F"
haftmann@34943: proof -
wenzelm@60500:   from \<open>F \<le># A\<close>
haftmann@34943:   show ?thesis
haftmann@34943:   proof (induct F)
haftmann@34943:     show "P {#}" by fact
haftmann@34943:   next
haftmann@34943:     fix x F
Mathias@60397:     assume P: "F \<le># A \<Longrightarrow> P F" and i: "F + {#x#} \<le># A"
haftmann@34943:     show "P (F + {#x#})"
haftmann@34943:     proof (rule insert)
haftmann@34943:       from i show "x \<in># A" by (auto dest: mset_le_insertD)
Mathias@60397:       from i have "F \<le># A" by (auto dest: mset_le_insertD)
haftmann@34943:       with P show "P F" .
haftmann@34943:     qed
haftmann@34943:   qed
haftmann@34943: qed
wenzelm@26145: 
wenzelm@17161: 
wenzelm@60500: subsection \<open>The fold combinator\<close>
huffman@48023: 
nipkow@59998: definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
huffman@48023: where
nipkow@60495:   "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
huffman@48023: 
wenzelm@60606: lemma fold_mset_empty [simp]: "fold_mset f s {#} = s"
nipkow@59998:   by (simp add: fold_mset_def)
huffman@48023: 
huffman@48023: context comp_fun_commute
huffman@48023: begin
huffman@48023: 
wenzelm@60606: lemma fold_mset_insert: "fold_mset f s (M + {#x#}) = f x (fold_mset f s M)"
haftmann@49822: proof -
haftmann@49822:   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
haftmann@49822:     by (fact comp_fun_commute_funpow)
haftmann@49822:   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
haftmann@49822:     by (fact comp_fun_commute_funpow)
haftmann@49822:   show ?thesis
nipkow@60495:   proof (cases "x \<in> set_mset M")
haftmann@49822:     case False
haftmann@49822:     then have *: "count (M + {#x#}) x = 1" by simp
nipkow@60495:     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_mset M) =
nipkow@60495:       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_mset M)"
haftmann@49822:       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
haftmann@49822:     with False * show ?thesis
nipkow@59998:       by (simp add: fold_mset_def del: count_union)
huffman@48023:   next
haftmann@49822:     case True
nipkow@60495:     def N \<equiv> "set_mset M - {x}"
nipkow@60495:     from N_def True have *: "set_mset M = insert x N" "x \<notin> N" "finite N" by auto
haftmann@49822:     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
haftmann@49822:       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
haftmann@49822:       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
nipkow@59998:     with * show ?thesis by (simp add: fold_mset_def del: count_union) simp
huffman@48023:   qed
huffman@48023: qed
huffman@48023: 
wenzelm@60606: corollary fold_mset_single [simp]: "fold_mset f s {#x#} = f x s"
haftmann@49822: proof -
nipkow@59998:   have "fold_mset f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
haftmann@49822:   then show ?thesis by simp
haftmann@49822: qed
huffman@48023: 
wenzelm@60606: lemma fold_mset_fun_left_comm: "f x (fold_mset f s M) = fold_mset f (f x s) M"
haftmann@49822:   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
huffman@48023: 
wenzelm@60606: lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
haftmann@49822: proof (induct M)
huffman@48023:   case empty then show ?case by simp
huffman@48023: next
haftmann@49822:   case (add M x)
haftmann@49822:   have "M + {#x#} + N = (M + N) + {#x#}"
haftmann@57514:     by (simp add: ac_simps)
haftmann@51548:   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
huffman@48023: qed
huffman@48023: 
huffman@48023: lemma fold_mset_fusion:
huffman@48023:   assumes "comp_fun_commute g"
wenzelm@60606:     and *: "\<And>x y. h (g x y) = f x (h y)"
wenzelm@60606:   shows "h (fold_mset g w A) = fold_mset f (h w) A"
huffman@48023: proof -
huffman@48023:   interpret comp_fun_commute g by (fact assms)
wenzelm@60606:   from * show ?thesis by (induct A) auto
huffman@48023: qed
huffman@48023: 
huffman@48023: end
huffman@48023: 
wenzelm@60500: text \<open>
huffman@48023:   A note on code generation: When defining some function containing a
nipkow@59998:   subterm @{term "fold_mset F"}, code generation is not automatic. When
wenzelm@61585:   interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the
nipkow@59998:   would be code thms for @{const fold_mset} become thms like
wenzelm@61585:   @{term "fold_mset F z {#} = z"} where \<open>F\<close> is not a pattern but
huffman@48023:   contains defined symbols, i.e.\ is not a code thm. Hence a separate
wenzelm@61585:   constant with its own code thms needs to be introduced for \<open>F\<close>. See the image operator below.
wenzelm@60500: \<close>
wenzelm@60500: 
wenzelm@60500: 
wenzelm@60500: subsection \<open>Image\<close>
huffman@48023: 
huffman@48023: definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
wenzelm@60607:   "image_mset f = fold_mset (plus \<circ> single \<circ> f) {#}"
wenzelm@60607: 
wenzelm@60607: lemma comp_fun_commute_mset_image: "comp_fun_commute (plus \<circ> single \<circ> f)"
haftmann@49823: proof
haftmann@57514: qed (simp add: ac_simps fun_eq_iff)
huffman@48023: 
huffman@48023: lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
haftmann@49823:   by (simp add: image_mset_def)
huffman@48023: 
huffman@48023: lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
haftmann@49823: proof -
wenzelm@60607:   interpret comp_fun_commute "plus \<circ> single \<circ> f"
haftmann@49823:     by (fact comp_fun_commute_mset_image)
haftmann@49823:   show ?thesis by (simp add: image_mset_def)
haftmann@49823: qed
huffman@48023: 
wenzelm@60606: lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
haftmann@49823: proof -
wenzelm@60607:   interpret comp_fun_commute "plus \<circ> single \<circ> f"
haftmann@49823:     by (fact comp_fun_commute_mset_image)
haftmann@57514:   show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
haftmann@49823: qed
haftmann@49823: 
wenzelm@60606: corollary image_mset_insert: "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
haftmann@49823:   by simp
huffman@48023: 
wenzelm@60606: lemma set_image_mset [simp]: "set_mset (image_mset f M) = image f (set_mset M)"
haftmann@49823:   by (induct M) simp_all
huffman@48040: 
wenzelm@60606: lemma size_image_mset [simp]: "size (image_mset f M) = size M"
haftmann@49823:   by (induct M) simp_all
huffman@48023: 
wenzelm@60606: lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
haftmann@49823:   by (cases M) auto
huffman@48023: 
huffman@48023: syntax
huffman@48023:   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
huffman@48023:       ("({#_/. _ :# _#})")
huffman@48023: translations
wenzelm@60606:   "{#e. x:#M#}" == "CONST image_mset (\<lambda>x. e) M"
huffman@48023: 
blanchet@59813: syntax (xsymbols)
blanchet@59813:   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
blanchet@59813:       ("({#_/. _ \<in># _#})")
blanchet@59813: translations
blanchet@59813:   "{#e. x \<in># M#}" == "CONST image_mset (\<lambda>x. e) M"
blanchet@59813: 
huffman@48023: syntax
blanchet@59813:   "_comprehension3_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
huffman@48023:       ("({#_/ | _ :# _./ _#})")
huffman@48023: translations
wenzelm@60606:   "{#e | x:#M. P#}" \<rightharpoonup> "{#e. x :# {# x:#M. P#}#}"
huffman@48023: 
blanchet@59813: syntax
blanchet@59813:   "_comprehension4_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
blanchet@59813:       ("({#_/ | _ \<in># _./ _#})")
blanchet@59813: translations
wenzelm@60606:   "{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}"
blanchet@59813: 
wenzelm@60500: text \<open>
wenzelm@60607:   This allows to write not just filters like @{term "{#x\<in>#M. x<c#}"}
wenzelm@60607:   but also images like @{term "{#x+x. x\<in>#M #}"} and @{term [source]
wenzelm@60607:   "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
wenzelm@60607:   @{term "{#x+x|x\<in>#M. x<c#}"}.
wenzelm@60500: \<close>
huffman@48023: 
nipkow@60495: lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
nipkow@60498: by (metis mem_set_mset_iff set_image_mset)
blanchet@59813: 
blanchet@55467: functor image_mset: image_mset
huffman@48023: proof -
huffman@48023:   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
huffman@48023:   proof
huffman@48023:     fix A
huffman@48023:     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
huffman@48023:       by (induct A) simp_all
huffman@48023:   qed
huffman@48023:   show "image_mset id = id"
huffman@48023:   proof
huffman@48023:     fix A
huffman@48023:     show "image_mset id A = id A"
huffman@48023:       by (induct A) simp_all
huffman@48023:   qed
huffman@48023: qed
huffman@48023: 
blanchet@59813: declare
blanchet@59813:   image_mset.id [simp]
blanchet@59813:   image_mset.identity [simp]
blanchet@59813: 
blanchet@59813: lemma image_mset_id[simp]: "image_mset id x = x"
blanchet@59813:   unfolding id_def by auto
blanchet@59813: 
blanchet@59813: 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:   by (induct M) auto
blanchet@59813: 
blanchet@59813: lemma image_mset_cong_pair:
blanchet@59813:   "(\<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:   by (metis image_mset_cong split_cong)
haftmann@49717: 
huffman@48023: 
wenzelm@60500: subsection \<open>Further conversions\<close>
haftmann@34943: 
nipkow@60515: primrec mset :: "'a list \<Rightarrow> 'a multiset" where
nipkow@60515:   "mset [] = {#}" |
nipkow@60515:   "mset (a # x) = mset x + {# a #}"
haftmann@34943: 
haftmann@37107: lemma in_multiset_in_set:
nipkow@60515:   "x \<in># mset xs \<longleftrightarrow> x \<in> set xs"
haftmann@37107:   by (induct xs) simp_all
haftmann@37107: 
nipkow@60515: lemma count_mset:
nipkow@60515:   "count (mset xs) x = length (filter (\<lambda>y. x = y) xs)"
haftmann@37107:   by (induct xs) simp_all
haftmann@37107: 
nipkow@60515: lemma mset_zero_iff[simp]: "(mset x = {#}) = (x = [])"
blanchet@59813:   by (induct x) auto
haftmann@34943: 
nipkow@60515: lemma mset_zero_iff_right[simp]: "({#} = mset x) = (x = [])"
haftmann@34943: by (induct x) auto
haftmann@34943: 
nipkow@60515: lemma set_mset_mset[simp]: "set_mset (mset x) = set x"
haftmann@34943: by (induct x) auto
haftmann@34943: 
wenzelm@60607: lemma mem_set_multiset_eq: "x \<in> set xs = (x \<in># mset xs)"
haftmann@34943: by (induct xs) auto
haftmann@34943: 
nipkow@60515: lemma size_mset [simp]: "size (mset xs) = length xs"
huffman@48012:   by (induct xs) simp_all
huffman@48012: 
wenzelm@60606: lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys"
haftmann@57514:   by (induct xs arbitrary: ys) (auto simp: ac_simps)
haftmann@34943: 
wenzelm@60607: lemma mset_filter: "mset (filter P xs) = {#x \<in># mset xs. P x #}"
haftmann@40303:   by (induct xs) simp_all
haftmann@40303: 
nipkow@60515: lemma mset_rev [simp]:
nipkow@60515:   "mset (rev xs) = mset xs"
haftmann@40950:   by (induct xs) simp_all
haftmann@40950: 
nipkow@60515: lemma surj_mset: "surj mset"
haftmann@34943: apply (unfold surj_def)
haftmann@34943: apply (rule allI)
haftmann@34943: apply (rule_tac M = y in multiset_induct)
haftmann@34943:  apply auto
haftmann@34943: apply (rule_tac x = "x # xa" in exI)
haftmann@34943: apply auto
haftmann@34943: done
haftmann@34943: 
nipkow@60515: lemma set_count_greater_0: "set x = {a. count (mset x) a > 0}"
haftmann@34943: by (induct x) auto
haftmann@34943: 
haftmann@34943: lemma distinct_count_atmost_1:
wenzelm@60606:   "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
wenzelm@60678:   apply (induct x, simp, rule iffI, simp_all)
wenzelm@60678:   subgoal for a b
wenzelm@60678:     apply (rule conjI)
wenzelm@60678:     apply (simp_all add: set_mset_mset [symmetric] del: set_mset_mset)
wenzelm@60678:     apply (erule_tac x = a in allE, simp)
wenzelm@60678:     apply clarify
wenzelm@60678:     apply (erule_tac x = aa in allE, simp)
wenzelm@60678:     done
wenzelm@60678:   done
haftmann@34943: 
wenzelm@60606: lemma mset_eq_setD: "mset xs = mset ys \<Longrightarrow> set xs = set ys"
nipkow@39302: by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
haftmann@34943: 
nipkow@60515: lemma set_eq_iff_mset_eq_distinct:
haftmann@34943:   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
nipkow@60515:     (set x = set y) = (mset x = mset y)"
nipkow@39302: by (auto simp: multiset_eq_iff distinct_count_atmost_1)
haftmann@34943: 
nipkow@60515: lemma set_eq_iff_mset_remdups_eq:
nipkow@60515:    "(set x = set y) = (mset (remdups x) = mset (remdups y))"
haftmann@34943: apply (rule iffI)
nipkow@60515: apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1])
haftmann@34943: apply (drule distinct_remdups [THEN distinct_remdups
nipkow@60515:       [THEN set_eq_iff_mset_eq_distinct [THEN iffD2]]])
haftmann@34943: apply simp
haftmann@34943: done
haftmann@34943: 
wenzelm@60606: lemma mset_compl_union [simp]: "mset [x\<leftarrow>xs. P x] + mset [x\<leftarrow>xs. \<not>P x] = mset xs"
haftmann@57514:   by (induct xs) (auto simp: ac_simps)
haftmann@34943: 
wenzelm@60607: lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
wenzelm@60678: proof (induct ls arbitrary: i)
wenzelm@60678:   case Nil
wenzelm@60678:   then show ?case by simp
wenzelm@60678: next
wenzelm@60678:   case Cons
wenzelm@60678:   then show ?case by (cases i) auto
wenzelm@60678: qed
haftmann@34943: 
wenzelm@60606: lemma mset_remove1[simp]: "mset (remove1 a xs) = mset xs - {#a#}"
wenzelm@60678:   by (induct xs) (auto simp add: multiset_eq_iff)
haftmann@34943: 
nipkow@60515: lemma mset_eq_length:
nipkow@60515:   assumes "mset xs = mset ys"
haftmann@37107:   shows "length xs = length ys"
nipkow@60515:   using assms by (metis size_mset)
nipkow@60515: 
nipkow@60515: lemma mset_eq_length_filter:
nipkow@60515:   assumes "mset xs = mset ys"
haftmann@39533:   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
nipkow@60515:   using assms by (metis count_mset)
haftmann@39533: 
haftmann@45989: lemma fold_multiset_equiv:
haftmann@45989:   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:     and equiv: "mset xs = mset ys"
haftmann@49822:   shows "List.fold f xs = List.fold f ys"
wenzelm@60606:   using f equiv [symmetric]
wenzelm@46921: proof (induct xs arbitrary: ys)
wenzelm@60678:   case Nil
wenzelm@60678:   then show ?case by simp
haftmann@45989: next
haftmann@45989:   case (Cons x xs)
wenzelm@60678:   then have *: "set ys = set (x # xs)"
wenzelm@60678:     by (blast dest: mset_eq_setD)
blanchet@58425:   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:     by (rule Cons.prems(1)) (simp_all add: *)
wenzelm@60678:   moreover from * have "x \<in> set ys"
wenzelm@60678:     by simp
wenzelm@60678:   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x"
wenzelm@60678:     by (fact fold_remove1_split)
wenzelm@60678:   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)"
wenzelm@60678:     by (auto intro: Cons.hyps)
haftmann@45989:   ultimately show ?case by simp
haftmann@45989: qed
haftmann@45989: 
wenzelm@60606: lemma mset_insort [simp]: "mset (insort x xs) = mset xs + {#x#}"
haftmann@51548:   by (induct xs) (simp_all add: ac_simps)
haftmann@51548: 
wenzelm@60606: lemma mset_map: "mset (map f xs) = image_mset f (mset xs)"
haftmann@51600:   by (induct xs) simp_all
haftmann@51600: 
nipkow@60513: definition mset_set :: "'a set \<Rightarrow> 'a multiset"
haftmann@51548: where
nipkow@60513:   "mset_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
nipkow@60513: 
nipkow@60513: interpretation mset_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
ballarin@61566: rewrites
nipkow@60513:   "folding.F (\<lambda>x M. {#x#} + M) {#} = mset_set"
haftmann@51548: proof -
wenzelm@60678:   interpret comp_fun_commute "\<lambda>x M. {#x#} + M"
wenzelm@60678:     by standard (simp add: fun_eq_iff ac_simps)
wenzelm@60678:   show "folding (\<lambda>x M. {#x#} + M)"
wenzelm@60678:     by standard (fact comp_fun_commute)
nipkow@60513:   from mset_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = mset_set" ..
haftmann@51548: qed
haftmann@51548: 
nipkow@60513: lemma count_mset_set [simp]:
nipkow@60513:   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (mset_set A) x = 1" (is "PROP ?P")
nipkow@60513:   "\<not> finite A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?Q")
nipkow@60513:   "x \<notin> A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?R")
haftmann@51600: proof -
wenzelm@60606:   have *: "count (mset_set A) x = 0" if "x \<notin> A" for A
wenzelm@60606:   proof (cases "finite A")
wenzelm@60606:     case False then show ?thesis by simp
wenzelm@60606:   next
wenzelm@60606:     case True from True \<open>x \<notin> A\<close> show ?thesis by (induct A) auto
wenzelm@60606:   qed
haftmann@51600:   then show "PROP ?P" "PROP ?Q" "PROP ?R"
haftmann@51600:   by (auto elim!: Set.set_insert)
wenzelm@61585: qed \<comment> \<open>TODO: maybe define @{const mset_set} also in terms of @{const Abs_multiset}\<close>
nipkow@60513: 
nipkow@60513: lemma elem_mset_set[simp, intro]: "finite A \<Longrightarrow> x \<in># mset_set A \<longleftrightarrow> x \<in> A"
blanchet@59813:   by (induct A rule: finite_induct) simp_all
blanchet@59813: 
haftmann@51548: context linorder
haftmann@51548: begin
haftmann@51548: 
haftmann@51548: definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
haftmann@51548: where
nipkow@59998:   "sorted_list_of_multiset M = fold_mset insort [] M"
haftmann@51548: 
haftmann@51548: lemma sorted_list_of_multiset_empty [simp]:
haftmann@51548:   "sorted_list_of_multiset {#} = []"
haftmann@51548:   by (simp add: sorted_list_of_multiset_def)
haftmann@51548: 
haftmann@51548: lemma sorted_list_of_multiset_singleton [simp]:
haftmann@51548:   "sorted_list_of_multiset {#x#} = [x]"
haftmann@51548: proof -
haftmann@51548:   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
haftmann@51548:   show ?thesis by (simp add: sorted_list_of_multiset_def)
haftmann@51548: qed
haftmann@51548: 
haftmann@51548: lemma sorted_list_of_multiset_insert [simp]:
haftmann@51548:   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
haftmann@51548: proof -
haftmann@51548:   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
haftmann@51548:   show ?thesis by (simp add: sorted_list_of_multiset_def)
haftmann@51548: qed
haftmann@51548: 
haftmann@51548: end
haftmann@51548: 
nipkow@60515: lemma mset_sorted_list_of_multiset [simp]:
nipkow@60515:   "mset (sorted_list_of_multiset M) = M"
nipkow@60513: by (induct M) simp_all
haftmann@51548: 
nipkow@60515: lemma sorted_list_of_multiset_mset [simp]:
nipkow@60515:   "sorted_list_of_multiset (mset xs) = sort xs"
nipkow@60513: by (induct xs) simp_all
nipkow@60513: 
nipkow@60513: lemma finite_set_mset_mset_set[simp]:
nipkow@60513:   "finite A \<Longrightarrow> set_mset (mset_set A) = A"
nipkow@60513: by (induct A rule: finite_induct) simp_all
nipkow@60513: 
nipkow@60513: lemma infinite_set_mset_mset_set:
nipkow@60513:   "\<not> finite A \<Longrightarrow> set_mset (mset_set A) = {}"
nipkow@60513: by simp
haftmann@51548: 
haftmann@51548: lemma set_sorted_list_of_multiset [simp]:
nipkow@60495:   "set (sorted_list_of_multiset M) = set_mset M"
nipkow@60513: by (induct M) (simp_all add: set_insort)
nipkow@60513: 
nipkow@60513: lemma sorted_list_of_mset_set [simp]:
nipkow@60513:   "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
nipkow@60513: by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
haftmann@51548: 
haftmann@51548: 
haftmann@60804: subsection \<open>Replicate operation\<close>
haftmann@60804: 
haftmann@60804: definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
haftmann@60804:   "replicate_mset n x = ((op + {#x#}) ^^ n) {#}"
haftmann@60804: 
haftmann@60804: lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
haftmann@60804:   unfolding replicate_mset_def by simp
haftmann@60804: 
haftmann@60804: lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}"
haftmann@60804:   unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
haftmann@60804: 
haftmann@60804: lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
haftmann@60804:   unfolding replicate_mset_def by (induct n) simp_all
haftmann@60804: 
haftmann@60804: lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
haftmann@60804:   unfolding replicate_mset_def by (induct n) simp_all
haftmann@60804: 
haftmann@60804: lemma set_mset_replicate_mset_subset[simp]: "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
haftmann@60804:   by (auto split: if_splits)
haftmann@60804: 
haftmann@60804: lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
haftmann@60804:   by (induct n, simp_all)
haftmann@60804: 
haftmann@60804: lemma count_le_replicate_mset_le: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<le># M"
haftmann@60804:   by (auto simp add: assms mset_less_eqI) (metis count_replicate_mset subseteq_mset_def)
haftmann@60804: 
haftmann@60804: lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
haftmann@60804:   by (induct D) simp_all
haftmann@60804: 
haftmann@61031: lemma replicate_count_mset_eq_filter_eq:
haftmann@61031:   "replicate (count (mset xs) k) k = filter (HOL.eq k) xs"
haftmann@61031:   by (induct xs) auto
haftmann@61031: 
haftmann@60804: 
wenzelm@60500: subsection \<open>Big operators\<close>
haftmann@51548: 
haftmann@51548: no_notation times (infixl "*" 70)
haftmann@51548: no_notation Groups.one ("1")
haftmann@51548: 
haftmann@51548: locale comm_monoid_mset = comm_monoid
haftmann@51548: begin
haftmann@51548: 
haftmann@51548: definition F :: "'a multiset \<Rightarrow> 'a"
wenzelm@60606:   where eq_fold: "F M = fold_mset f 1 M"
wenzelm@60606: 
wenzelm@60606: lemma empty [simp]: "F {#} = 1"
haftmann@51548:   by (simp add: eq_fold)
haftmann@51548: 
wenzelm@60678: lemma singleton [simp]: "F {#x#} = x"
haftmann@51548: proof -
haftmann@51548:   interpret comp_fun_commute
wenzelm@60678:     by standard (simp add: fun_eq_iff left_commute)
haftmann@51548:   show ?thesis by (simp add: eq_fold)
haftmann@51548: qed
haftmann@51548: 
wenzelm@60606: lemma union [simp]: "F (M + N) = F M * F N"
haftmann@51548: proof -
haftmann@51548:   interpret comp_fun_commute f
wenzelm@60678:     by standard (simp add: fun_eq_iff left_commute)
wenzelm@60678:   show ?thesis
wenzelm@60678:     by (induct N) (simp_all add: left_commute eq_fold)
haftmann@51548: qed
haftmann@51548: 
haftmann@51548: end
haftmann@51548: 
wenzelm@61076: lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
wenzelm@60678:   by standard (simp add: add_ac comp_def)
blanchet@59813: 
blanchet@59813: declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]
blanchet@59813: 
nipkow@59998: 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:   by (induct NN) auto
blanchet@59813: 
haftmann@51548: notation times (infixl "*" 70)
haftmann@51548: notation Groups.one ("1")
haftmann@51548: 
haftmann@54868: context comm_monoid_add
haftmann@54868: begin
haftmann@54868: 
haftmann@54868: definition msetsum :: "'a multiset \<Rightarrow> 'a"
wenzelm@60606:   where "msetsum = comm_monoid_mset.F plus 0"
haftmann@51548: 
haftmann@54868: sublocale msetsum!: comm_monoid_mset plus 0
ballarin@61566:   rewrites "comm_monoid_mset.F plus 0 = msetsum"
haftmann@51548: proof -
haftmann@51548:   show "comm_monoid_mset plus 0" ..
haftmann@51548:   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
haftmann@51548: qed
haftmann@51548: 
haftmann@60804: lemma (in semiring_1) msetsum_replicate_mset [simp]:
haftmann@60804:   "msetsum (replicate_mset n a) = of_nat n * a"
haftmann@60804:   by (induct n) (simp_all add: algebra_simps)
haftmann@60804: 
haftmann@51548: lemma setsum_unfold_msetsum:
nipkow@60513:   "setsum f A = msetsum (image_mset f (mset_set A))"
haftmann@51548:   by (cases "finite A") (induct A rule: finite_induct, simp_all)
haftmann@51548: 
haftmann@51548: end
haftmann@51548: 
blanchet@59813: lemma msetsum_diff:
wenzelm@61076:   fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset"
Mathias@60397:   shows "N \<le># M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N"
Mathias@60397:   by (metis add_diff_cancel_right' msetsum.union subset_mset.diff_add)
blanchet@59813: 
nipkow@59949: lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)"
nipkow@59949: proof (induct M)
nipkow@59949:   case empty then show ?case by simp
nipkow@59949: next
nipkow@59949:   case (add M x) then show ?case
nipkow@60495:     by (cases "x \<in> set_mset M")
nipkow@60495:       (simp_all del: mem_set_mset_iff add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb, simp)
nipkow@59949: qed
nipkow@59949: 
nipkow@59949: 
blanchet@59813: abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset" where
blanchet@59813:   "Union_mset MM \<equiv> msetsum MM"
blanchet@59813: 
blanchet@59813: notation (xsymbols) Union_mset ("\<Union>#_" [900] 900)
blanchet@59813: 
nipkow@60495: lemma set_mset_Union_mset[simp]: "set_mset (\<Union># MM) = (\<Union>M \<in> set_mset MM. set_mset M)"
blanchet@59813:   by (induct MM) auto
blanchet@59813: 
blanchet@59813: lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
blanchet@59813:   by (induct MM) auto
blanchet@59813: 
haftmann@51548: syntax
blanchet@58425:   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
haftmann@51548:       ("(3SUM _:#_. _)" [0, 51, 10] 10)
haftmann@51548: syntax (xsymbols)
blanchet@58425:   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
haftmann@57518:       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
haftmann@51548: translations
haftmann@57518:   "SUM i :# A. b" == "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
haftmann@51548: 
haftmann@54868: context comm_monoid_mult
haftmann@54868: begin
haftmann@54868: 
haftmann@54868: definition msetprod :: "'a multiset \<Rightarrow> 'a"
wenzelm@60606:   where "msetprod = comm_monoid_mset.F times 1"
haftmann@54868: 
haftmann@54868: sublocale msetprod!: comm_monoid_mset times 1
ballarin@61566:   rewrites "comm_monoid_mset.F times 1 = msetprod"
haftmann@51548: proof -
haftmann@51548:   show "comm_monoid_mset times 1" ..
wenzelm@60606:   show "comm_monoid_mset.F times 1 = msetprod" using msetprod_def ..
haftmann@51548: qed
haftmann@51548: 
haftmann@51548: lemma msetprod_empty:
haftmann@51548:   "msetprod {#} = 1"
haftmann@51548:   by (fact msetprod.empty)
haftmann@51548: 
haftmann@51548: lemma msetprod_singleton:
haftmann@51548:   "msetprod {#x#} = x"
haftmann@51548:   by (fact msetprod.singleton)
haftmann@51548: 
haftmann@51548: lemma msetprod_Un:
blanchet@58425:   "msetprod (A + B) = msetprod A * msetprod B"
haftmann@51548:   by (fact msetprod.union)
haftmann@51548: 
haftmann@60804: lemma msetprod_replicate_mset [simp]:
haftmann@60804:   "msetprod (replicate_mset n a) = a ^ n"
haftmann@60804:   by (induct n) (simp_all add: ac_simps)
haftmann@60804: 
haftmann@51548: lemma setprod_unfold_msetprod:
nipkow@60513:   "setprod f A = msetprod (image_mset f (mset_set A))"
haftmann@51548:   by (cases "finite A") (induct A rule: finite_induct, simp_all)
haftmann@51548: 
haftmann@51548: lemma msetprod_multiplicity:
nipkow@60495:   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_mset M)"
nipkow@59998:   by (simp add: fold_mset_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
haftmann@51548: 
haftmann@51548: end
haftmann@51548: 
haftmann@51548: syntax
blanchet@58425:   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
haftmann@51548:       ("(3PROD _:#_. _)" [0, 51, 10] 10)
haftmann@51548: syntax (xsymbols)
blanchet@58425:   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
haftmann@51548:       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
haftmann@51548: translations
haftmann@57518:   "PROD i :# A. b" == "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
haftmann@51548: 
haftmann@51548: lemma (in comm_semiring_1) dvd_msetprod:
haftmann@51548:   assumes "x \<in># A"
haftmann@51548:   shows "x dvd msetprod A"
haftmann@51548: proof -
haftmann@51548:   from assms have "A = (A - {#x#}) + {#x#}" by simp
haftmann@51548:   then obtain B where "A = B + {#x#}" ..
haftmann@51548:   then show ?thesis by simp
haftmann@51548: qed
haftmann@51548: 
haftmann@51548: 
wenzelm@60500: subsection \<open>Alternative representations\<close>
wenzelm@60500: 
wenzelm@60500: subsubsection \<open>Lists\<close>
haftmann@51548: 
haftmann@39533: context linorder
haftmann@39533: begin
haftmann@39533: 
nipkow@60515: lemma mset_insort [simp]:
nipkow@60515:   "mset (insort_key k x xs) = {#x#} + mset xs"
haftmann@37107:   by (induct xs) (simp_all add: ac_simps)
blanchet@58425: 
nipkow@60515: lemma mset_sort [simp]:
nipkow@60515:   "mset (sort_key k xs) = mset xs"
haftmann@37107:   by (induct xs) (simp_all add: ac_simps)
haftmann@37107: 
wenzelm@60500: text \<open>
haftmann@34943:   This lemma shows which properties suffice to show that a function
wenzelm@61585:   \<open>f\<close> with \<open>f xs = ys\<close> behaves like sort.
wenzelm@60500: \<close>
haftmann@37074: 
haftmann@39533: lemma properties_for_sort_key:
nipkow@60515:   assumes "mset ys = mset xs"
wenzelm@60606:     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:     and "sorted (map f ys)"
haftmann@39533:   shows "sort_key f xs = ys"
wenzelm@60606:   using assms
wenzelm@46921: proof (induct xs arbitrary: ys)
haftmann@34943:   case Nil then show ?case by simp
haftmann@34943: next
haftmann@34943:   case (Cons x xs)
haftmann@39533:   from Cons.prems(2) have
haftmann@40305:     "\<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:     by (simp add: filter_remove1)
haftmann@39533:   with Cons.prems have "sort_key f xs = remove1 x ys"
haftmann@39533:     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
haftmann@39533:   moreover from Cons.prems have "x \<in> set ys"
haftmann@39533:     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
haftmann@39533:   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
haftmann@34943: qed
haftmann@34943: 
haftmann@39533: lemma properties_for_sort:
nipkow@60515:   assumes multiset: "mset ys = mset xs"
wenzelm@60606:     and "sorted ys"
haftmann@39533:   shows "sort xs = ys"
haftmann@39533: proof (rule properties_for_sort_key)
nipkow@60515:   from multiset show "mset ys = mset xs" .
wenzelm@60500:   from \<open>sorted ys\<close> show "sorted (map (\<lambda>x. x) ys)" by simp
wenzelm@60678:   from multiset have "length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" for k
nipkow@60515:     by (rule mset_eq_length_filter)
wenzelm@60678:   then have "replicate (length (filter (\<lambda>y. k = y) ys)) k =
wenzelm@60678:     replicate (length (filter (\<lambda>x. k = x) xs)) k" for k
haftmann@39533:     by simp
wenzelm@60678:   then show "k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" for k
haftmann@39533:     by (simp add: replicate_length_filter)
haftmann@39533: qed
haftmann@39533: 
haftmann@61031: lemma sort_key_inj_key_eq:
haftmann@61031:   assumes mset_equal: "mset xs = mset ys"
haftmann@61031:     and "inj_on f (set xs)"
haftmann@61031:     and "sorted (map f ys)"
haftmann@61031:   shows "sort_key f xs = ys"
haftmann@61031: proof (rule properties_for_sort_key)
haftmann@61031:   from mset_equal
haftmann@61031:   show "mset ys = mset xs" by simp
wenzelm@61188:   from \<open>sorted (map f ys)\<close>
haftmann@61031:   show "sorted (map f ys)" .
haftmann@61031:   show "[x\<leftarrow>ys . f k = f x] = [x\<leftarrow>xs . f k = f x]" if "k \<in> set ys" for k
haftmann@61031:   proof -
haftmann@61031:     from mset_equal
haftmann@61031:     have set_equal: "set xs = set ys" by (rule mset_eq_setD)
haftmann@61031:     with that have "insert k (set ys) = set ys" by auto
wenzelm@61188:     with \<open>inj_on f (set xs)\<close> have inj: "inj_on f (insert k (set ys))"
haftmann@61031:       by (simp add: set_equal)
haftmann@61031:     from inj have "[x\<leftarrow>ys . f k = f x] = filter (HOL.eq k) ys"
haftmann@61031:       by (auto intro!: inj_on_filter_key_eq)
haftmann@61031:     also have "\<dots> = replicate (count (mset ys) k) k"
haftmann@61031:       by (simp add: replicate_count_mset_eq_filter_eq)
haftmann@61031:     also have "\<dots> = replicate (count (mset xs) k) k"
haftmann@61031:       using mset_equal by simp
haftmann@61031:     also have "\<dots> = filter (HOL.eq k) xs"
haftmann@61031:       by (simp add: replicate_count_mset_eq_filter_eq)
haftmann@61031:     also have "\<dots> = [x\<leftarrow>xs . f k = f x]"
haftmann@61031:       using inj by (auto intro!: inj_on_filter_key_eq [symmetric] simp add: set_equal)
haftmann@61031:     finally show ?thesis .
haftmann@61031:   qed
haftmann@61031: qed
haftmann@61031: 
haftmann@61031: lemma sort_key_eq_sort_key:
haftmann@61031:   assumes "mset xs = mset ys"
haftmann@61031:     and "inj_on f (set xs)"
haftmann@61031:   shows "sort_key f xs = sort_key f ys"
haftmann@61031:   by (rule sort_key_inj_key_eq) (simp_all add: assms)
haftmann@61031: 
haftmann@40303: lemma sort_key_by_quicksort:
haftmann@40303:   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
haftmann@40303:     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
haftmann@40303:     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
haftmann@40303: proof (rule properties_for_sort_key)
nipkow@60515:   show "mset ?rhs = mset ?lhs"
nipkow@60515:     by (rule multiset_eqI) (auto simp add: mset_filter)
haftmann@40303:   show "sorted (map f ?rhs)"
haftmann@40303:     by (auto simp add: sorted_append intro: sorted_map_same)
haftmann@40303: next
haftmann@40305:   fix l
haftmann@40305:   assume "l \<in> set ?rhs"
haftmann@40346:   let ?pivot = "f (xs ! (length xs div 2))"
haftmann@40346:   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
haftmann@40306:   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
haftmann@40305:     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
haftmann@40346:   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
haftmann@40346:   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:   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
haftmann@40346:     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
haftmann@40346:   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
haftmann@40306:   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
haftmann@40305:   proof (cases "f l" ?pivot rule: linorder_cases)
wenzelm@46730:     case less
wenzelm@46730:     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
wenzelm@46730:     with less show ?thesis
haftmann@40346:       by (simp add: filter_sort [symmetric] ** ***)
haftmann@40305:   next
haftmann@40306:     case equal then show ?thesis
haftmann@40346:       by (simp add: * less_le)
haftmann@40305:   next
wenzelm@46730:     case greater
wenzelm@46730:     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
wenzelm@46730:     with greater show ?thesis
haftmann@40346:       by (simp add: filter_sort [symmetric] ** ***)
haftmann@40306:   qed
haftmann@40303: qed
haftmann@40303: 
haftmann@40303: lemma sort_by_quicksort:
haftmann@40303:   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
haftmann@40303:     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
haftmann@40303:     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
haftmann@40303:   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
haftmann@40303: 
wenzelm@60500: text \<open>A stable parametrized quicksort\<close>
haftmann@40347: 
haftmann@40347: definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
haftmann@40347:   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
haftmann@40347: 
haftmann@40347: lemma part_code [code]:
haftmann@40347:   "part f pivot [] = ([], [], [])"
haftmann@40347:   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
haftmann@40347:      if x' < pivot then (x # lts, eqs, gts)
haftmann@40347:      else if x' > pivot then (lts, eqs, x # gts)
haftmann@40347:      else (lts, x # eqs, gts))"
haftmann@40347:   by (auto simp add: part_def Let_def split_def)
haftmann@40347: 
haftmann@40347: lemma sort_key_by_quicksort_code [code]:
wenzelm@60606:   "sort_key f xs =
wenzelm@60606:     (case xs of
wenzelm@60606:       [] \<Rightarrow> []
haftmann@40347:     | [x] \<Rightarrow> xs
haftmann@40347:     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
wenzelm@60606:     | _ \<Rightarrow>
wenzelm@60606:         let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
wenzelm@60606:         in sort_key f lts @ eqs @ sort_key f gts)"
haftmann@40347: proof (cases xs)
haftmann@40347:   case Nil then show ?thesis by simp
haftmann@40347: next
wenzelm@46921:   case (Cons _ ys) note hyps = Cons show ?thesis
wenzelm@46921:   proof (cases ys)
haftmann@40347:     case Nil with hyps show ?thesis by simp
haftmann@40347:   next
wenzelm@46921:     case (Cons _ zs) note hyps = hyps Cons show ?thesis
wenzelm@46921:     proof (cases zs)
haftmann@40347:       case Nil with hyps show ?thesis by auto
haftmann@40347:     next
blanchet@58425:       case Cons
haftmann@40347:       from sort_key_by_quicksort [of f xs]
haftmann@40347:       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
haftmann@40347:         in sort_key f lts @ eqs @ sort_key f gts)"
haftmann@40347:       by (simp only: split_def Let_def part_def fst_conv snd_conv)
haftmann@40347:       with hyps Cons show ?thesis by (simp only: list.cases)
haftmann@40347:     qed
haftmann@40347:   qed
haftmann@40347: qed
haftmann@40347: 
haftmann@39533: end
haftmann@39533: 
haftmann@40347: hide_const (open) part
haftmann@40347: 
nipkow@60515: lemma mset_remdups_le: "mset (remdups xs) \<le># mset xs"
Mathias@60397:   by (induct xs) (auto intro: subset_mset.order_trans)
haftmann@34943: 
nipkow@60515: lemma mset_update:
nipkow@60515:   "i < length ls \<Longrightarrow> mset (ls[i := v]) = mset ls - {#ls ! i#} + {#v#}"
haftmann@34943: proof (induct ls arbitrary: i)
haftmann@34943:   case Nil then show ?case by simp
haftmann@34943: next
haftmann@34943:   case (Cons x xs)
haftmann@34943:   show ?case
haftmann@34943:   proof (cases i)
haftmann@34943:     case 0 then show ?thesis by simp
haftmann@34943:   next
haftmann@34943:     case (Suc i')
haftmann@34943:     with Cons show ?thesis
haftmann@34943:       apply simp
haftmann@57512:       apply (subst add.assoc)
haftmann@57512:       apply (subst add.commute [of "{#v#}" "{#x#}"])
haftmann@57512:       apply (subst add.assoc [symmetric])
haftmann@34943:       apply simp
haftmann@34943:       apply (rule mset_le_multiset_union_diff_commute)
nipkow@60515:       apply (simp add: mset_le_single nth_mem_mset)
haftmann@34943:       done
haftmann@34943:   qed
haftmann@34943: qed
haftmann@34943: 
nipkow@60515: lemma mset_swap:
haftmann@34943:   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
nipkow@60515:     mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
nipkow@60515:   by (cases "i = j") (simp_all add: mset_update nth_mem_mset)
haftmann@34943: 
haftmann@34943: 
wenzelm@60500: subsection \<open>The multiset order\<close>
wenzelm@60500: 
wenzelm@60500: subsubsection \<open>Well-foundedness\<close>
wenzelm@10249: 
wenzelm@60606: definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
haftmann@37765:   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
wenzelm@60607:       (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
wenzelm@60606: 
wenzelm@60606: definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
haftmann@37765:   "mult r = (mult1 r)\<^sup>+"
wenzelm@10249: 
berghofe@23751: lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
nipkow@26178: by (simp add: mult1_def)
wenzelm@10249: 
wenzelm@60608: lemma less_add:
wenzelm@60608:   assumes mult1: "(N, M0 + {#a#}) \<in> mult1 r"
wenzelm@60608:   shows
wenzelm@60608:     "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
wenzelm@60608:      (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@60608: proof -
wenzelm@60607:   let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
nipkow@11464:   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
wenzelm@60608:   obtain a' M0' K where M0: "M0 + {#a#} = M0' + {#a'#}"
wenzelm@60608:     and N: "N = M0' + K"
wenzelm@60608:     and r: "?r K a'"
wenzelm@60608:     using mult1 unfolding mult1_def by auto
wenzelm@60608:   show ?thesis (is "?case1 \<or> ?case2")
wenzelm@60606:   proof -
wenzelm@60606:     from M0 consider "M0 = M0'" "a = a'"
wenzelm@60606:       | K' where "M0 = K' + {#a'#}" "M0' = K' + {#a#}"
wenzelm@60606:       by atomize_elim (simp only: add_eq_conv_ex)
wenzelm@18258:     then show ?thesis
wenzelm@60606:     proof cases
wenzelm@60606:       case 1
nipkow@11464:       with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@60606:       then have ?case2 ..
wenzelm@60606:       then show ?thesis ..
wenzelm@10249:     next
wenzelm@60606:       case 2
wenzelm@60606:       from N 2(2) have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
wenzelm@60606:       with r 2(1) have "?R (K' + K) M0" by blast
wenzelm@60608:       with n have ?case1 by (simp add: mult1_def)
wenzelm@60606:       then show ?thesis ..
wenzelm@10249:     qed
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
wenzelm@60608: lemma all_accessible:
wenzelm@60608:   assumes "wf r"
wenzelm@60608:   shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)"
wenzelm@10249: proof
wenzelm@10249:   let ?R = "mult1 r"
haftmann@54295:   let ?W = "Wellfounded.acc ?R"
wenzelm@10249:   {
wenzelm@10249:     fix M M0 a
berghofe@23751:     assume M0: "M0 \<in> ?W"
wenzelm@60606:       and wf_hyp: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
wenzelm@60606:       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W"
berghofe@23751:     have "M0 + {#a#} \<in> ?W"
berghofe@23751:     proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249:       fix N
berghofe@23751:       assume "(N, M0 + {#a#}) \<in> ?R"
wenzelm@60608:       then consider M where "(M, M0) \<in> ?R" "N = M + {#a#}"
wenzelm@60608:         | K where "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" "N = M0 + K"
wenzelm@60608:         by atomize_elim (rule less_add)
berghofe@23751:       then show "N \<in> ?W"
wenzelm@60608:       proof cases
wenzelm@60608:         case 1
wenzelm@60606:         from acc_hyp have "(M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W" ..
wenzelm@60500:         from this and \<open>(M, M0) \<in> ?R\<close> have "M + {#a#} \<in> ?W" ..
wenzelm@60608:         then show "N \<in> ?W" by (simp only: \<open>N = M + {#a#}\<close>)
wenzelm@10249:       next
wenzelm@60608:         case 2
wenzelm@60608:         from this(1) have "M0 + K \<in> ?W"
wenzelm@10249:         proof (induct K)
wenzelm@18730:           case empty
berghofe@23751:           from M0 show "M0 + {#} \<in> ?W" by simp
wenzelm@18730:         next
wenzelm@18730:           case (add K x)
berghofe@23751:           from add.prems have "(x, a) \<in> r" by simp
berghofe@23751:           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
berghofe@23751:           moreover from add have "M0 + K \<in> ?W" by simp
berghofe@23751:           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
haftmann@57512:           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
wenzelm@10249:         qed
wenzelm@60608:         then show "N \<in> ?W" by (simp only: 2(2))
wenzelm@10249:       qed
wenzelm@10249:     qed
wenzelm@10249:   } note tedious_reasoning = this
wenzelm@10249: 
wenzelm@60608:   show "M \<in> ?W" for M
wenzelm@10249:   proof (induct M)
berghofe@23751:     show "{#} \<in> ?W"
wenzelm@10249:     proof (rule accI)
berghofe@23751:       fix b assume "(b, {#}) \<in> ?R"
berghofe@23751:       with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249:     qed
wenzelm@10249: 
berghofe@23751:     fix M a assume "M \<in> ?W"
wenzelm@60608:     from \<open>wf r\<close> have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:     proof induct
wenzelm@10249:       fix a
wenzelm@60606:       assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751:       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:       proof
berghofe@23751:         fix M assume "M \<in> ?W"
berghofe@23751:         then show "M + {#a#} \<in> ?W"
wenzelm@23373:           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
wenzelm@10249:       qed
wenzelm@10249:     qed
wenzelm@60500:     from this and \<open>M \<in> ?W\<close> show "M + {#a#} \<in> ?W" ..
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
wenzelm@60606: theorem wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)"
nipkow@26178: by (rule acc_wfI) (rule all_accessible)
wenzelm@10249: 
wenzelm@60606: theorem wf_mult: "wf r \<Longrightarrow> wf (mult r)"
nipkow@26178: unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@60500: subsubsection \<open>Closure-free presentation\<close>
wenzelm@60500: 
wenzelm@60500: text \<open>One direction.\<close>
wenzelm@10249: 
wenzelm@10249: lemma mult_implies_one_step:
wenzelm@60606:   "trans r \<Longrightarrow> (M, N) \<in> mult r \<Longrightarrow>
nipkow@11464:     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
nipkow@60495:     (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)"
nipkow@60495: apply (unfold mult_def mult1_def set_mset_def)
nipkow@26178: apply (erule converse_trancl_induct, clarify)
nipkow@26178:  apply (rule_tac x = M0 in exI, simp, clarify)
wenzelm@60607: apply (case_tac "a \<in># K")
nipkow@26178:  apply (rule_tac x = I in exI)
nipkow@26178:  apply (simp (no_asm))
nipkow@26178:  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
haftmann@57512:  apply (simp (no_asm_simp) add: add.assoc [symmetric])
wenzelm@59807:  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
nipkow@26178:  apply (simp add: diff_union_single_conv)
nipkow@26178:  apply (simp (no_asm_use) add: trans_def)
nipkow@26178:  apply blast
wenzelm@60607: apply (subgoal_tac "a \<in># I")
nipkow@26178:  apply (rule_tac x = "I - {#a#}" in exI)
nipkow@26178:  apply (rule_tac x = "J + {#a#}" in exI)
nipkow@26178:  apply (rule_tac x = "K + Ka" in exI)
nipkow@26178:  apply (rule conjI)
nipkow@39302:   apply (simp add: multiset_eq_iff split: nat_diff_split)
nipkow@26178:  apply (rule conjI)
wenzelm@59807:   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
nipkow@39302:   apply (simp add: multiset_eq_iff split: nat_diff_split)
nipkow@26178:  apply (simp (no_asm_use) add: trans_def)
nipkow@26178:  apply blast
wenzelm@60607: apply (subgoal_tac "a \<in># (M0 + {#a#})")
nipkow@26178:  apply simp
nipkow@26178: apply (simp (no_asm))
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma one_step_implies_mult_aux:
wenzelm@60678:   "\<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:     \<longrightarrow> (I + K, I + J) \<in> mult r"
wenzelm@60678: apply (induct n)
wenzelm@60678:  apply auto
nipkow@26178: apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow@26178: apply (rename_tac "J'", simp)
nipkow@26178: apply (erule notE, auto)
nipkow@26178: apply (case_tac "J' = {#}")
nipkow@26178:  apply (simp add: mult_def)
nipkow@26178:  apply (rule r_into_trancl)
nipkow@60495:  apply (simp add: mult1_def set_mset_def, blast)
wenzelm@60500: txt \<open>Now we know @{term "J' \<noteq> {#}"}.\<close>
nipkow@26178: apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@60495: apply (erule_tac P = "\<forall>k \<in> set_mset K. P k" for P in rev_mp)
nipkow@26178: apply (erule ssubst)
nipkow@26178: apply (simp add: Ball_def, auto)
nipkow@26178: apply (subgoal_tac
wenzelm@60607:   "((I + {# x \<in># K. (x, a) \<in> r #}) + {# x \<in># K. (x, a) \<notin> r #},
wenzelm@60607:     (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow@26178:  prefer 2
nipkow@26178:  apply force
haftmann@57512: apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
nipkow@26178: apply (erule trancl_trans)
nipkow@26178: apply (rule r_into_trancl)
nipkow@60495: apply (simp add: mult1_def set_mset_def)
nipkow@26178: apply (rule_tac x = a in exI)
nipkow@26178: apply (rule_tac x = "I + J'" in exI)
haftmann@57514: apply (simp add: ac_simps)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma one_step_implies_mult:
wenzelm@60606:   "trans r \<Longrightarrow> J \<noteq> {#} \<Longrightarrow> \<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r
wenzelm@60606:     \<Longrightarrow> (I + K, I + J) \<in> mult r"
nipkow@26178: using one_step_implies_mult_aux by blast
wenzelm@10249: 
wenzelm@10249: 
wenzelm@60500: subsubsection \<open>Partial-order properties\<close>
wenzelm@10249: 
wenzelm@61076: definition less_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "#<#" 50) where
blanchet@59958:   "M' #<# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
blanchet@59958: 
wenzelm@61076: definition le_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "#<=#" 50) where
blanchet@59958:   "M' #<=# M \<longleftrightarrow> M' #<# M \<or> M' = M"
blanchet@59958: 
blanchet@59958: notation (xsymbols) less_multiset (infix "#\<subset>#" 50)
blanchet@59958: notation (xsymbols) le_multiset (infix "#\<subseteq>#" 50)
wenzelm@10249: 
haftmann@35268: interpretation multiset_order: order le_multiset less_multiset
haftmann@35268: proof -
wenzelm@60606:   have irrefl: "\<not> M #\<subset># M" for M :: "'a multiset"
haftmann@35268:   proof
blanchet@59958:     assume "M #\<subset># M"
haftmann@35268:     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
haftmann@35268:     have "trans {(x'::'a, x). x' < x}"
haftmann@35268:       by (rule transI) simp
haftmann@35268:     moreover note MM
haftmann@35268:     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
nipkow@60495:       \<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:       by (rule mult_implies_one_step)
haftmann@35268:     then obtain I J K where "M = I + J" and "M = I + K"
nipkow@60495:       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:     then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. k < j" by auto
nipkow@60495:     have "finite (set_mset K)" by simp
wenzelm@60678:     moreover note **
nipkow@60495:     ultimately have "set_mset K = {}"
haftmann@35268:       by (induct rule: finite_induct) (auto intro: order_less_trans)
wenzelm@60678:     with * show False by simp
haftmann@35268:   qed
wenzelm@60678:   have trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># N" for K M N :: "'a multiset"
haftmann@35268:     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@46921:   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
wenzelm@60678:     by standard (auto simp add: le_multiset_def irrefl dest: trans)
haftmann@35268: qed
wenzelm@10249: 
wenzelm@60678: lemma mult_less_irrefl [elim!]:
wenzelm@60678:   fixes M :: "'a::order multiset"
wenzelm@60678:   shows "M #\<subset># M \<Longrightarrow> R"
wenzelm@46730:   by simp
haftmann@26567: 
wenzelm@10249: 
wenzelm@60500: subsubsection \<open>Monotonicity of multiset union\<close>
wenzelm@10249: 
wenzelm@60606: lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r"
nipkow@26178: apply (unfold mult1_def)
nipkow@26178: apply auto
nipkow@26178: apply (rule_tac x = a in exI)
nipkow@26178: apply (rule_tac x = "C + M0" in exI)
haftmann@57512: apply (simp add: add.assoc)
nipkow@26178: done
wenzelm@10249: 
wenzelm@60606: lemma union_less_mono2: "B #\<subset># D \<Longrightarrow> C + B #\<subset># C + (D::'a::order multiset)"
nipkow@26178: apply (unfold less_multiset_def mult_def)
nipkow@26178: apply (erule trancl_induct)
noschinl@40249:  apply (blast intro: mult1_union)
noschinl@40249: apply (blast intro: mult1_union trancl_trans)
nipkow@26178: done
wenzelm@10249: 
wenzelm@60606: lemma union_less_mono1: "B #\<subset># D \<Longrightarrow> B + C #\<subset># D + (C::'a::order multiset)"
haftmann@57512: apply (subst add.commute [of B C])
haftmann@57512: apply (subst add.commute [of D C])
nipkow@26178: apply (erule union_less_mono2)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma union_less_mono:
wenzelm@60606:   fixes A B C D :: "'a::order multiset"
wenzelm@60606:   shows "A #\<subset># C \<Longrightarrow> B #\<subset># D \<Longrightarrow> A + B #\<subset># C + D"
haftmann@35268:   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
wenzelm@10249: 
haftmann@35268: interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
wenzelm@60678:   by standard (auto simp add: le_multiset_def intro: union_less_mono2)
wenzelm@26145: 
paulson@15072: 
wenzelm@60500: subsubsection \<open>Termination proofs with multiset orders\<close>
krauss@29125: 
krauss@29125: lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
krauss@29125:   and multi_member_this: "x \<in># {# x #} + XS"
krauss@29125:   and multi_member_last: "x \<in># {# x #}"
krauss@29125:   by auto
krauss@29125: 
krauss@29125: definition "ms_strict = mult pair_less"
haftmann@37765: definition "ms_weak = ms_strict \<union> Id"
krauss@29125: 
krauss@29125: lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
krauss@29125: unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
krauss@29125: by (auto intro: wf_mult1 wf_trancl simp: mult_def)
krauss@29125: 
krauss@29125: lemma smsI:
nipkow@60495:   "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
krauss@29125:   unfolding ms_strict_def
krauss@29125: by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
krauss@29125: 
krauss@29125: lemma wmsI:
nipkow@60495:   "(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#}
krauss@29125:   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
krauss@29125: unfolding ms_weak_def ms_strict_def
krauss@29125: by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
krauss@29125: 
krauss@29125: inductive pw_leq
krauss@29125: where
krauss@29125:   pw_leq_empty: "pw_leq {#} {#}"
krauss@29125: | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
krauss@29125: 
krauss@29125: lemma pw_leq_lstep:
krauss@29125:   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
krauss@29125: by (drule pw_leq_step) (rule pw_leq_empty, simp)
krauss@29125: 
krauss@29125: lemma pw_leq_split:
krauss@29125:   assumes "pw_leq X Y"
nipkow@60495:   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:   using assms
wenzelm@60606: proof induct
krauss@29125:   case pw_leq_empty thus ?case by auto
krauss@29125: next
krauss@29125:   case (pw_leq_step x y X Y)
krauss@29125:   then obtain A B Z where
blanchet@58425:     [simp]: "X = A + Z" "Y = B + Z"
nipkow@60495:       and 1[simp]: "(set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
krauss@29125:     by auto
wenzelm@60606:   from pw_leq_step consider "x = y" | "(x, y) \<in> pair_less"
krauss@29125:     unfolding pair_leq_def by auto
krauss@29125:   thus ?case
wenzelm@60606:   proof cases
wenzelm@60606:     case [simp]: 1
wenzelm@60606:     have "{#x#} + X = A + ({#y#}+Z) \<and> {#y#} + Y = B + ({#y#}+Z) \<and>
wenzelm@60606:       ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
haftmann@57514:       by (auto simp: ac_simps)
wenzelm@60606:     thus ?thesis by blast
krauss@29125:   next
wenzelm@60606:     case 2
krauss@29125:     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
krauss@29125:     have "{#x#} + X = ?A' + Z"
krauss@29125:       "{#y#} + Y = ?B' + Z"
haftmann@57514:       by (auto simp add: ac_simps)
blanchet@58425:     moreover have
nipkow@60495:       "(set_mset ?A', set_mset ?B') \<in> max_strict"
wenzelm@60606:       using 1 2 unfolding max_strict_def
krauss@29125:       by (auto elim!: max_ext.cases)
krauss@29125:     ultimately show ?thesis by blast
krauss@29125:   qed
krauss@29125: qed
krauss@29125: 
blanchet@58425: lemma
krauss@29125:   assumes pwleq: "pw_leq Z Z'"
nipkow@60495:   shows ms_strictI: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
wenzelm@60606:     and ms_weakI1:  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
wenzelm@60606:     and ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
krauss@29125: proof -
blanchet@58425:   from pw_leq_split[OF pwleq]
krauss@29125:   obtain A' B' Z''
krauss@29125:     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
nipkow@60495:     and mx_or_empty: "(set_mset A', set_mset B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
krauss@29125:     by blast
krauss@29125:   {
nipkow@60495:     assume max: "(set_mset A, set_mset B) \<in> max_strict"
krauss@29125:     from mx_or_empty
krauss@29125:     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
krauss@29125:     proof
nipkow@60495:       assume max': "(set_mset A', set_mset B') \<in> max_strict"
nipkow@60495:       with max have "(set_mset (A + A'), set_mset (B + B')) \<in> max_strict"
krauss@29125:         by (auto simp: max_strict_def intro: max_ext_additive)
blanchet@58425:       thus ?thesis by (rule smsI)
krauss@29125:     next
krauss@29125:       assume [simp]: "A' = {#} \<and> B' = {#}"
krauss@29125:       show ?thesis by (rule smsI) (auto intro: max)
krauss@29125:     qed
wenzelm@60606:     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add: ac_simps)
krauss@29125:     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
krauss@29125:   }
krauss@29125:   from mx_or_empty
krauss@29125:   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
haftmann@57514:   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
krauss@29125: qed
krauss@29125: 
nipkow@39301: lemma empty_neutral: "{#} + x = x" "x + {#} = x"
krauss@29125: and nonempty_plus: "{# x #} + rs \<noteq> {#}"
krauss@29125: and nonempty_single: "{# x #} \<noteq> {#}"
krauss@29125: by auto
krauss@29125: 
wenzelm@60500: setup \<open>
wenzelm@60606:   let
wenzelm@60606:     fun msetT T = Type (@{type_name multiset}, [T]);
wenzelm@60606: 
wenzelm@60606:     fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
wenzelm@60606:       | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
wenzelm@60606:       | mk_mset T (x :: xs) =
wenzelm@60606:             Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
wenzelm@60606:                   mk_mset T [x] $ mk_mset T xs
wenzelm@60606: 
wenzelm@60752:     fun mset_member_tac ctxt m i =
wenzelm@60606:       if m <= 0 then
wenzelm@60752:         resolve_tac ctxt @{thms multi_member_this} i ORELSE
wenzelm@60752:         resolve_tac ctxt @{thms multi_member_last} i
wenzelm@60606:       else
wenzelm@60752:         resolve_tac ctxt @{thms multi_member_skip} i THEN mset_member_tac ctxt (m - 1) i
wenzelm@60752: 
wenzelm@60752:     fun mset_nonempty_tac ctxt =
wenzelm@60752:       resolve_tac ctxt @{thms nonempty_plus} ORELSE'
wenzelm@60752:       resolve_tac ctxt @{thms nonempty_single}
krauss@29125: 
wenzelm@60606:     fun regroup_munion_conv ctxt =
wenzelm@60606:       Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
wenzelm@60606:         (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
wenzelm@60606: 
wenzelm@60752:     fun unfold_pwleq_tac ctxt i =
wenzelm@60752:       (resolve_tac ctxt @{thms pw_leq_step} i THEN (fn st => unfold_pwleq_tac ctxt (i + 1) st))
wenzelm@60752:         ORELSE (resolve_tac ctxt @{thms pw_leq_lstep} i)
wenzelm@60752:         ORELSE (resolve_tac ctxt @{thms pw_leq_empty} i)
wenzelm@60606: 
wenzelm@60606:     val set_mset_simps = [@{thm set_mset_empty}, @{thm set_mset_single}, @{thm set_mset_union},
wenzelm@60606:                         @{thm Un_insert_left}, @{thm Un_empty_left}]
wenzelm@60606:   in
wenzelm@60606:     ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
wenzelm@60606:     {
wenzelm@60606:       msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
wenzelm@60606:       mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
wenzelm@60606:       mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_mset_simps,
wenzelm@60606:       smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
wenzelm@60752:       reduction_pair = @{thm ms_reduction_pair}
wenzelm@60606:     })
wenzelm@60606:   end
wenzelm@60500: \<close>
wenzelm@60500: 
wenzelm@60500: 
wenzelm@60500: subsection \<open>Legacy theorem bindings\<close>
haftmann@34943: 
nipkow@39302: lemmas multi_count_eq = multiset_eq_iff [symmetric]
haftmann@34943: 
haftmann@34943: lemma union_commute: "M + N = N + (M::'a multiset)"
haftmann@57512:   by (fact add.commute)
haftmann@34943: 
haftmann@34943: lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
haftmann@57512:   by (fact add.assoc)
haftmann@34943: 
haftmann@34943: lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
haftmann@57512:   by (fact add.left_commute)
haftmann@34943: 
haftmann@34943: lemmas union_ac = union_assoc union_commute union_lcomm
haftmann@34943: 
haftmann@34943: lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
haftmann@34943:   by (fact add_right_cancel)
haftmann@34943: 
haftmann@34943: lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
haftmann@34943:   by (fact add_left_cancel)
haftmann@34943: 
haftmann@34943: lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
haftmann@59557:   by (fact add_left_imp_eq)
haftmann@34943: 
Mathias@60397: lemma mset_less_trans: "(M::'a multiset) <# K \<Longrightarrow> K <# N \<Longrightarrow> M <# N"
Mathias@60397:   by (fact subset_mset.less_trans)
haftmann@35268: 
haftmann@35268: lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
Mathias@60397:   by (fact subset_mset.inf.commute)
haftmann@35268: 
haftmann@35268: lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
Mathias@60397:   by (fact subset_mset.inf.assoc [symmetric])
haftmann@35268: 
haftmann@35268: lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
Mathias@60397:   by (fact subset_mset.inf.left_commute)
haftmann@35268: 
haftmann@35268: lemmas multiset_inter_ac =
haftmann@35268:   multiset_inter_commute
haftmann@35268:   multiset_inter_assoc
haftmann@35268:   multiset_inter_left_commute
haftmann@35268: 
wenzelm@60606: lemma mult_less_not_refl: "\<not> M #\<subset># (M::'a::order multiset)"
haftmann@35268:   by (fact multiset_order.less_irrefl)
haftmann@35268: 
wenzelm@60606: lemma mult_less_trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># (N::'a::order multiset)"
haftmann@35268:   by (fact multiset_order.less_trans)
blanchet@58425: 
wenzelm@60606: lemma mult_less_not_sym: "M #\<subset># N \<Longrightarrow> \<not> N #\<subset># (M::'a::order multiset)"
haftmann@35268:   by (fact multiset_order.less_not_sym)
haftmann@35268: 
wenzelm@60606: lemma mult_less_asym: "M #\<subset># N \<Longrightarrow> (\<not> P \<Longrightarrow> N #\<subset># (M::'a::order multiset)) \<Longrightarrow> P"
haftmann@35268:   by (fact multiset_order.less_asym)
haftmann@34943: 
wenzelm@60500: declaration \<open>
wenzelm@60606:   let
wenzelm@60606:     fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) (Const _ $ t') =
wenzelm@60606:           let
wenzelm@60606:             val (maybe_opt, ps) =
wenzelm@60606:               Nitpick_Model.dest_plain_fun t'
wenzelm@60606:               ||> op ~~
wenzelm@60606:               ||> map (apsnd (snd o HOLogic.dest_number))
wenzelm@60606:             fun elems_for t =
wenzelm@60606:               (case AList.lookup (op =) ps t of
wenzelm@60606:                 SOME n => replicate n t
wenzelm@60606:               | NONE => [Const (maybe_name, elem_T --> elem_T) $ t])
wenzelm@60606:           in
wenzelm@60606:             (case maps elems_for (all_values elem_T) @
blanchet@61333:                  (if maybe_opt then [Const (Nitpick_Model.unrep_mixfix (), elem_T)] else []) of
wenzelm@60606:               [] => Const (@{const_name zero_class.zero}, T)
wenzelm@60606:             | ts =>
wenzelm@60606:                 foldl1 (fn (t1, t2) =>
wenzelm@60606:                     Const (@{const_name plus_class.plus}, T --> T --> T) $ t1 $ t2)
wenzelm@60606:                   (map (curry (op $) (Const (@{const_name single}, elem_T --> T))) ts))
wenzelm@60606:           end
wenzelm@60606:       | multiset_postproc _ _ _ _ t = t
wenzelm@60606:   in Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} multiset_postproc end
wenzelm@60500: \<close>
wenzelm@60500: 
wenzelm@60500: 
wenzelm@60500: subsection \<open>Naive implementation using lists\<close>
haftmann@51600: 
nipkow@60515: code_datatype mset
haftmann@51600: 
wenzelm@60606: lemma [code]: "{#} = mset []"
haftmann@51600:   by simp
haftmann@51600: 
wenzelm@60606: lemma [code]: "{#x#} = mset [x]"
haftmann@51600:   by simp
haftmann@51600: 
wenzelm@60606: lemma union_code [code]: "mset xs + mset ys = mset (xs @ ys)"
haftmann@51600:   by simp
haftmann@51600: 
wenzelm@60606: lemma [code]: "image_mset f (mset xs) = mset (map f xs)"
nipkow@60515:   by (simp add: mset_map)
haftmann@51600: 
wenzelm@60606: lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"
nipkow@60515:   by (simp add: mset_filter)
haftmann@51600: 
wenzelm@60606: lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)"
haftmann@51600:   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
haftmann@51600: 
haftmann@51600: lemma [code]:
nipkow@60515:   "mset xs #\<inter> mset ys =
nipkow@60515:     mset (snd (fold (\<lambda>x (ys, zs).
haftmann@51600:       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
haftmann@51600: proof -
nipkow@60515:   have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs).
haftmann@51600:     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
nipkow@60515:       (mset xs #\<inter> mset ys) + mset zs"
haftmann@51623:     by (induct xs arbitrary: ys)
haftmann@51623:       (auto simp add: mem_set_multiset_eq inter_add_right1 inter_add_right2 ac_simps)
haftmann@51623:   then show ?thesis by simp
haftmann@51623: qed
haftmann@51623: 
haftmann@51623: lemma [code]:
nipkow@60515:   "mset xs #\<union> mset ys =
haftmann@61424:     mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
haftmann@51623: proof -
haftmann@61424:   have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
nipkow@60515:       (mset xs #\<union> mset ys) + mset zs"
haftmann@51623:     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
haftmann@51600:   then show ?thesis by simp
haftmann@51600: qed
haftmann@51600: 
blanchet@59813: declare in_multiset_in_set [code_unfold]
haftmann@51600: 
wenzelm@60606: lemma [code]: "count (mset xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
haftmann@51600: proof -
nipkow@60515:   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (mset xs) x + n"
haftmann@51600:     by (induct xs) simp_all
haftmann@51600:   then show ?thesis by simp
haftmann@51600: qed
haftmann@51600: 
nipkow@60515: declare set_mset_mset [code]
nipkow@60515: 
nipkow@60515: declare sorted_list_of_multiset_mset [code]
haftmann@51600: 
wenzelm@61585: lemma [code]: \<comment> \<open>not very efficient, but representation-ignorant!\<close>
nipkow@60515:   "mset_set A = mset (sorted_list_of_set A)"
haftmann@51600:   apply (cases "finite A")
haftmann@51600:   apply simp_all
haftmann@51600:   apply (induct A rule: finite_induct)
blanchet@59813:   apply (simp_all add: add.commute)
haftmann@51600:   done
haftmann@51600: 
nipkow@60515: declare size_mset [code]
haftmann@51600: 
blanchet@58425: fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
nipkow@55808:   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
blanchet@58425: | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
nipkow@55808:      None \<Rightarrow> None
nipkow@55808:    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
nipkow@55808: 
nipkow@60515: lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<le># mset ys) \<and>
nipkow@60515:   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> mset xs <# mset ys) \<and>
nipkow@60515:   (ms_lesseq_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
nipkow@55808: proof (induct xs arbitrary: ys)
nipkow@55808:   case (Nil ys)
nipkow@55808:   show ?case by (auto simp: mset_less_empty_nonempty)
nipkow@55808: next
nipkow@55808:   case (Cons x xs ys)
nipkow@55808:   show ?case
nipkow@55808:   proof (cases "List.extract (op = x) ys")
nipkow@55808:     case None
nipkow@55808:     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
nipkow@55808:     {
nipkow@60515:       assume "mset (x # xs) \<le># mset ys"
nipkow@60495:       from set_mset_mono[OF this] x have False by simp
nipkow@55808:     } note nle = this
nipkow@55808:     moreover
nipkow@55808:     {
nipkow@60515:       assume "mset (x # xs) <# mset ys"
nipkow@60515:       hence "mset (x # xs) \<le># mset ys" by auto
nipkow@55808:       from nle[OF this] have False .
nipkow@55808:     }
nipkow@55808:     ultimately show ?thesis using None by auto
nipkow@55808:   next
nipkow@55808:     case (Some res)
nipkow@55808:     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
nipkow@55808:     note Some = Some[unfolded res]
nipkow@55808:     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
nipkow@60515:     hence id: "mset ys = mset (ys1 @ ys2) + {#x#}"
nipkow@55808:       by (auto simp: ac_simps)
nipkow@55808:     show ?thesis unfolding ms_lesseq_impl.simps
nipkow@55808:       unfolding Some option.simps split
nipkow@55808:       unfolding id
nipkow@55808:       using Cons[of "ys1 @ ys2"]
Mathias@60397:       unfolding subset_mset_def subseteq_mset_def by auto
nipkow@55808:   qed
nipkow@55808: qed
nipkow@55808: 
nipkow@60515: lemma [code]: "mset xs \<le># mset ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
nipkow@55808:   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
nipkow@55808: 
nipkow@60515: lemma [code]: "mset xs <# mset ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
nipkow@55808:   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
haftmann@51600: 
haftmann@51600: instantiation multiset :: (equal) equal
haftmann@51600: begin
haftmann@51600: 
haftmann@51600: definition
nipkow@55808:   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
nipkow@60515: lemma [code]: "HOL.equal (mset xs) (mset ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
nipkow@55808:   unfolding equal_multiset_def
nipkow@55808:   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
haftmann@51600: 
haftmann@51600: instance
wenzelm@60678:   by standard (simp add: equal_multiset_def)
wenzelm@60678: 
blanchet@37169: end
haftmann@49388: 
wenzelm@60606: lemma [code]: "msetsum (mset xs) = listsum xs"
haftmann@51600:   by (induct xs) (simp_all add: add.commute)
haftmann@51600: 
wenzelm@60606: lemma [code]: "msetprod (mset xs) = fold times xs 1"
haftmann@51600: proof -
nipkow@60515:   have "\<And>x. fold times xs x = msetprod (mset xs) * x"
haftmann@51600:     by (induct xs) (simp_all add: mult.assoc)
haftmann@51600:   then show ?thesis by simp
haftmann@51600: qed
haftmann@51600: 
wenzelm@60500: text \<open>
haftmann@51600:   Exercise for the casual reader: add implementations for @{const le_multiset}
haftmann@51600:   and @{const less_multiset} (multiset order).
wenzelm@60500: \<close>
wenzelm@60500: 
wenzelm@60500: text \<open>Quickcheck generators\<close>
haftmann@51600: 
haftmann@51600: definition (in term_syntax)
wenzelm@61076:   msetify :: "'a::typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
haftmann@51600:     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
nipkow@60515:   [code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {\<cdot>} xs"
haftmann@51600: 
haftmann@51600: notation fcomp (infixl "\<circ>>" 60)
haftmann@51600: notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@51600: 
haftmann@51600: instantiation multiset :: (random) random
haftmann@51600: begin
haftmann@51600: 
haftmann@51600: definition
haftmann@51600:   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
haftmann@51600: 
haftmann@51600: instance ..
haftmann@51600: 
haftmann@51600: end
haftmann@51600: 
haftmann@51600: no_notation fcomp (infixl "\<circ>>" 60)
haftmann@51600: no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@51600: 
haftmann@51600: instantiation multiset :: (full_exhaustive) full_exhaustive
haftmann@51600: begin
haftmann@51600: 
haftmann@51600: 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: where
haftmann@51600:   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
haftmann@51600: 
haftmann@51600: instance ..
haftmann@51600: 
haftmann@51600: end
haftmann@51600: 
haftmann@51600: hide_const (open) msetify
haftmann@51600: 
blanchet@55129: 
wenzelm@60500: subsection \<open>BNF setup\<close>
blanchet@55129: 
blanchet@57966: definition rel_mset where
nipkow@60515:   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
nipkow@60515: 
nipkow@60515: lemma mset_zip_take_Cons_drop_twice:
blanchet@57966:   assumes "length xs = length ys" "j \<le> length xs"
nipkow@60515:   shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
nipkow@60515:     mset (zip xs ys) + {#(x, y)#}"
wenzelm@60606:   using assms
blanchet@57966: proof (induct xs ys arbitrary: x y j rule: list_induct2)
blanchet@57966:   case Nil
blanchet@57966:   thus ?case
blanchet@57966:     by simp
blanchet@55129: next
blanchet@57966:   case (Cons x xs y ys)
blanchet@57966:   thus ?case
blanchet@57966:   proof (cases "j = 0")
blanchet@57966:     case True
blanchet@57966:     thus ?thesis
blanchet@57966:       by simp
blanchet@55129:   next
blanchet@57966:     case False
blanchet@57966:     then obtain k where k: "j = Suc k"
wenzelm@60678:       by (cases j) simp
blanchet@57966:     hence "k \<le> length xs"
blanchet@57966:       using Cons.prems by auto
nipkow@60515:     hence "mset (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
nipkow@60515:       mset (zip xs ys) + {#(x, y)#}"
blanchet@57966:       by (rule Cons.hyps(2))
blanchet@57966:     thus ?thesis
blanchet@57966:       unfolding k by (auto simp: add.commute union_lcomm)
blanchet@58425:   qed
blanchet@55129: qed
blanchet@55129: 
nipkow@60515: lemma ex_mset_zip_left:
nipkow@60515:   assumes "length xs = length ys" "mset xs' = mset xs"
nipkow@60515:   shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
blanchet@58425: using assms
blanchet@57966: proof (induct xs ys arbitrary: xs' rule: list_induct2)
blanchet@57966:   case Nil
blanchet@57966:   thus ?case
blanchet@57966:     by auto
blanchet@57966: next
blanchet@57966:   case (Cons x xs y ys xs')
blanchet@57966:   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
nipkow@60515:     by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD)
blanchet@58425: 
blanchet@58425:   def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
nipkow@60515:   have "mset xs' = {#x#} + mset xsa"
blanchet@57966:     unfolding xsa_def using j_len nth_j
nipkow@58247:     by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
nipkow@60515:       mset.simps(2) union_code add.commute)
nipkow@60515:   hence ms_x: "mset xsa = mset xs"
nipkow@60515:     by (metis Cons.prems add.commute add_right_imp_eq mset.simps(2))
blanchet@57966:   then obtain ysa where
nipkow@60515:     len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)"
blanchet@57966:     using Cons.hyps(2) by blast
blanchet@57966: 
blanchet@57966:   def ys' \<equiv> "take j ysa @ y # drop j ysa"
blanchet@57966:   have xs': "xs' = take j xsa @ x # drop j xsa"
blanchet@57966:     using ms_x j_len nth_j Cons.prems xsa_def
nipkow@58247:     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
nipkow@60515:       length_drop size_mset)
blanchet@57966:   have j_len': "j \<le> length xsa"
blanchet@57966:     using j_len xs' xsa_def
blanchet@57966:     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
blanchet@57966:   have "length ys' = length xs'"
blanchet@57966:     unfolding ys'_def using Cons.prems len_a ms_x
nipkow@60515:     by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length)
nipkow@60515:   moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))"
blanchet@57966:     unfolding xs' ys'_def
nipkow@60515:     by (rule trans[OF mset_zip_take_Cons_drop_twice])
blanchet@57966:       (auto simp: len_a ms_a j_len' add.commute)
blanchet@57966:   ultimately show ?case
blanchet@57966:     by blast
blanchet@55129: qed
blanchet@55129: 
blanchet@57966: lemma list_all2_reorder_left_invariance:
nipkow@60515:   assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs"
nipkow@60515:   shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
blanchet@57966: proof -
blanchet@57966:   have len: "length xs = length ys"
blanchet@57966:     using rel list_all2_conv_all_nth by auto
blanchet@57966:   obtain ys' where
nipkow@60515:     len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)"
nipkow@60515:     using len ms_x by (metis ex_mset_zip_left)
blanchet@57966:   have "list_all2 R xs' ys'"
nipkow@60515:     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD)
nipkow@60515:   moreover have "mset ys' = mset ys"
nipkow@60515:     using len len' ms_xy map_snd_zip mset_map by metis
blanchet@57966:   ultimately show ?thesis
blanchet@57966:     by blast
blanchet@55129: qed
blanchet@55129: 
nipkow@60515: lemma ex_mset: "\<exists>xs. mset xs = X"
nipkow@60515:   by (induct X) (simp, metis mset.simps(2))
blanchet@55129: 
blanchet@55129: bnf "'a multiset"
blanchet@57966:   map: image_mset
nipkow@60495:   sets: set_mset
blanchet@55129:   bd: natLeq
blanchet@55129:   wits: "{#}"
blanchet@57966:   rel: rel_mset
blanchet@57966: proof -
blanchet@57966:   show "image_mset id = id"
blanchet@57966:     by (rule image_mset.id)
wenzelm@60606:   show "image_mset (g \<circ> f) = image_mset g \<circ> image_mset f" for f g
blanchet@59813:     unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
wenzelm@60606:   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:     by (induct X) (simp_all (no_asm),
nipkow@60495:       metis One_nat_def Un_iff count_single mem_set_mset_iff set_mset_union zero_less_Suc)
wenzelm@60606:   show "set_mset \<circ> image_mset f = op ` f \<circ> set_mset" for f
blanchet@57966:     by auto
blanchet@57966:   show "card_order natLeq"
blanchet@57966:     by (rule natLeq_card_order)
blanchet@57966:   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
blanchet@57966:     by (rule natLeq_cinfinite)
wenzelm@60606:   show "ordLeq3 (card_of (set_mset X)) natLeq" for X
blanchet@57966:     by transfer
blanchet@57966:       (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
wenzelm@60606:   show "rel_mset R OO rel_mset S \<le> rel_mset (R OO S)" for R S
blanchet@57966:     unfolding rel_mset_def[abs_def] OO_def
blanchet@57966:     apply clarify
wenzelm@60678:     subgoal for X Z Y xs ys' ys zs
wenzelm@60678:       apply (drule list_all2_reorder_left_invariance [where xs = ys' and ys = zs and xs' = ys])
wenzelm@60678:       apply (auto intro: list_all2_trans)
wenzelm@60678:       done
wenzelm@60606:     done
wenzelm@60606:   show "rel_mset R =
nipkow@60495:     (BNF_Def.Grp {x. set_mset x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
wenzelm@60606:     BNF_Def.Grp {x. set_mset x \<subseteq> {(x, y). R x y}} (image_mset snd)" for R
blanchet@57966:     unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
blanchet@57966:     apply (rule ext)+
blanchet@57966:     apply auto
nipkow@60515:      apply (rule_tac x = "mset (zip xs ys)" in exI; auto)
nipkow@60515:         apply (metis list_all2_lengthD map_fst_zip mset_map)
blanchet@57966:        apply (auto simp: list_all2_iff)[1]
nipkow@60515:       apply (metis list_all2_lengthD map_snd_zip mset_map)
blanchet@57966:      apply (auto simp: list_all2_iff)[1]
blanchet@57966:     apply (rename_tac XY)
nipkow@60515:     apply (cut_tac X = XY in ex_mset)
blanchet@57966:     apply (erule exE)
blanchet@57966:     apply (rename_tac xys)
blanchet@57966:     apply (rule_tac x = "map fst xys" in exI)
nipkow@60515:     apply (auto simp: mset_map)
blanchet@57966:     apply (rule_tac x = "map snd xys" in exI)
nipkow@60515:     apply (auto simp: mset_map list_all2I subset_eq zip_map_fst_snd)
wenzelm@59997:     done
wenzelm@60606:   show "z \<in> set_mset {#} \<Longrightarrow> False" for z
blanchet@57966:     by auto
blanchet@57966: qed
blanchet@57966: 
wenzelm@60606: inductive rel_mset'
wenzelm@60606: where
blanchet@57966:   Zero[intro]: "rel_mset' R {#} {#}"
blanchet@57966: | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
blanchet@57966: 
blanchet@57966: lemma rel_mset_Zero: "rel_mset R {#} {#}"
blanchet@57966: unfolding rel_mset_def Grp_def by auto
blanchet@55129: 
blanchet@55129: declare multiset.count[simp]
blanchet@55129: declare Abs_multiset_inverse[simp]
blanchet@55129: declare multiset.count_inverse[simp]
blanchet@55129: declare union_preserves_multiset[simp]
blanchet@55129: 
blanchet@57966: lemma rel_mset_Plus:
wenzelm@60606:   assumes ab: "R a b"
wenzelm@60606:     and MN: "rel_mset R M N"
wenzelm@60606:   shows "rel_mset R (M + {#a#}) (N + {#b#})"
wenzelm@60606: proof -
wenzelm@60606:   have "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
wenzelm@60606:     image_mset snd y + {#b#} = image_mset snd ya \<and>
wenzelm@60606:     set_mset ya \<subseteq> {(x, y). R x y}"
wenzelm@60606:     if "R a b" and "set_mset y \<subseteq> {(x, y). R x y}" for y
wenzelm@60606:     using that by (intro exI[of _ "y + {#(a,b)#}"]) auto
blanchet@55129:   thus ?thesis
blanchet@55129:   using assms
blanchet@57966:   unfolding multiset.rel_compp_Grp Grp_def by blast
blanchet@55129: qed
blanchet@55129: 
wenzelm@60606: lemma rel_mset'_imp_rel_mset: "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
wenzelm@60678:   by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus)
blanchet@57966: 
wenzelm@60606: lemma rel_mset_size: "rel_mset R M N \<Longrightarrow> size M = size N"
wenzelm@60678:   unfolding multiset.rel_compp_Grp Grp_def by auto
blanchet@55129: 
blanchet@55129: lemma multiset_induct2[case_names empty addL addR]:
wenzelm@60678:   assumes empty: "P {#} {#}"
wenzelm@60678:     and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
wenzelm@60678:     and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
wenzelm@60678:   shows "P M N"
blanchet@55129: apply(induct N rule: multiset_induct)
blanchet@55129:   apply(induct M rule: multiset_induct, rule empty, erule addL)
blanchet@55129:   apply(induct M rule: multiset_induct, erule addR, erule addR)
blanchet@55129: done
blanchet@55129: 
nipkow@59949: lemma multiset_induct2_size[consumes 1, case_names empty add]:
wenzelm@60606:   assumes c: "size M = size N"
wenzelm@60606:     and empty: "P {#} {#}"
wenzelm@60606:     and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
wenzelm@60606:   shows "P M N"
wenzelm@60678:   using c
wenzelm@60678: proof (induct M arbitrary: N rule: measure_induct_rule[of size])
wenzelm@60606:   case (less M)
wenzelm@60606:   show ?case
blanchet@55129:   proof(cases "M = {#}")
blanchet@55129:     case True hence "N = {#}" using less.prems by auto
blanchet@55129:     thus ?thesis using True empty by auto
blanchet@55129:   next
blanchet@55129:     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
blanchet@55129:     have "N \<noteq> {#}" using False less.prems by auto
blanchet@55129:     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
nipkow@59949:     have "size M1 = size N1" using less.prems unfolding M N by auto
blanchet@55129:     thus ?thesis using M N less.hyps add by auto
blanchet@55129:   qed
blanchet@55129: qed
blanchet@55129: 
blanchet@55129: lemma msed_map_invL:
wenzelm@60606:   assumes "image_mset f (M + {#a#}) = N"
wenzelm@60606:   shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
wenzelm@60606: proof -
blanchet@55129:   have "f a \<in># N"
wenzelm@60606:     using assms multiset.set_map[of f "M + {#a#}"] by auto
blanchet@55129:   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
blanchet@57966:   have "image_mset f M = N1" using assms unfolding N by simp
blanchet@55129:   thus ?thesis using N by blast
blanchet@55129: qed
blanchet@55129: 
blanchet@55129: lemma msed_map_invR:
wenzelm@60606:   assumes "image_mset f M = N + {#b#}"
wenzelm@60606:   shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
wenzelm@60606: proof -
blanchet@55129:   obtain a where a: "a \<in># M" and fa: "f a = b"
wenzelm@60606:     using multiset.set_map[of f M] unfolding assms
wenzelm@60606:     by (metis image_iff mem_set_mset_iff union_single_eq_member)
blanchet@55129:   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
blanchet@57966:   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
blanchet@55129:   thus ?thesis using M fa by blast
blanchet@55129: qed
blanchet@55129: 
blanchet@55129: lemma msed_rel_invL:
wenzelm@60606:   assumes "rel_mset R (M + {#a#}) N"
wenzelm@60606:   shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
wenzelm@60606: proof -
blanchet@57966:   obtain K where KM: "image_mset fst K = M + {#a#}"
wenzelm@60606:     and KN: "image_mset snd K = N" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
wenzelm@60606:     using assms
wenzelm@60606:     unfolding multiset.rel_compp_Grp Grp_def by auto
blanchet@55129:   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
wenzelm@60606:     and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
blanchet@57966:   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
wenzelm@60606:     using msed_map_invL[OF KN[unfolded K]] by auto
blanchet@55129:   have Rab: "R a (snd ab)" using sK a unfolding K by auto
blanchet@57966:   have "rel_mset R M N1" using sK K1M K1N1
wenzelm@60606:     unfolding K multiset.rel_compp_Grp Grp_def by auto
blanchet@55129:   thus ?thesis using N Rab by auto
blanchet@55129: qed
blanchet@55129: 
blanchet@55129: lemma msed_rel_invR:
wenzelm@60606:   assumes "rel_mset R M (N + {#b#})"
wenzelm@60606:   shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
wenzelm@60606: proof -
blanchet@57966:   obtain K where KN: "image_mset snd K = N + {#b#}"
wenzelm@60606:     and KM: "image_mset fst K = M" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
wenzelm@60606:     using assms
wenzelm@60606:     unfolding multiset.rel_compp_Grp Grp_def by auto
blanchet@55129:   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
wenzelm@60606:     and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
blanchet@57966:   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
wenzelm@60606:     using msed_map_invL[OF KM[unfolded K]] by auto
blanchet@55129:   have Rab: "R (fst ab) b" using sK b unfolding K by auto
blanchet@57966:   have "rel_mset R M1 N" using sK K1N K1M1
wenzelm@60606:     unfolding K multiset.rel_compp_Grp Grp_def by auto
blanchet@55129:   thus ?thesis using M Rab by auto
blanchet@55129: qed
blanchet@55129: 
blanchet@57966: lemma rel_mset_imp_rel_mset':
wenzelm@60606:   assumes "rel_mset R M N"
wenzelm@60606:   shows "rel_mset' R M N"
nipkow@59949: using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size])
blanchet@55129:   case (less M)
nipkow@59949:   have c: "size M = size N" using rel_mset_size[OF less.prems] .
blanchet@55129:   show ?case
blanchet@55129:   proof(cases "M = {#}")
blanchet@55129:     case True hence "N = {#}" using c by simp
blanchet@57966:     thus ?thesis using True rel_mset'.Zero by auto
blanchet@55129:   next
blanchet@55129:     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
blanchet@57966:     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
wenzelm@60606:       using msed_rel_invL[OF less.prems[unfolded M]] by auto
blanchet@57966:     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
blanchet@57966:     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
blanchet@55129:   qed
blanchet@55129: qed
blanchet@55129: 
wenzelm@60606: lemma rel_mset_rel_mset': "rel_mset R M N = rel_mset' R M N"
wenzelm@60678:   using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
blanchet@57966: 
hoelzl@60613: text \<open>The main end product for @{const rel_mset}: inductive characterization:\<close>
wenzelm@61337: lemmas rel_mset_induct[case_names empty add, induct pred: rel_mset] =
wenzelm@60606:   rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
blanchet@55129: 
blanchet@56656: 
wenzelm@60500: subsection \<open>Size setup\<close>
blanchet@56656: 
blanchet@57966: lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
wenzelm@60678:   apply (rule ext)
wenzelm@60678:   subgoal for x by (induct x) auto
wenzelm@60678:   done
blanchet@56656: 
wenzelm@60500: setup \<open>
wenzelm@60606:   BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
wenzelm@60606:     @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
wenzelm@60606:       size_union}
wenzelm@60606:     @{thms multiset_size_o_map}
wenzelm@60500: \<close>
blanchet@56656: 
blanchet@56656: hide_const (open) wcount
blanchet@56656: 
blanchet@55129: end