
wenzelm@10249: (*  Title:      HOL/Library/Multiset.thy
wenzelm@10249:     ID:         $Id$
paulson@15072:     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
wenzelm@10249: *)
wenzelm@10249: 
wenzelm@14706: header {* Multisets *}
wenzelm@10249: 
nipkow@15131: theory Multiset
krauss@19564: imports Main
nipkow@15131: begin
wenzelm@10249: 
wenzelm@10249: subsection {* The type of multisets *}
wenzelm@10249: 
nipkow@25162: typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
wenzelm@10249: proof
nipkow@11464:   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
wenzelm@10249: qed
wenzelm@10249: 
wenzelm@10249: lemmas multiset_typedef [simp] =
wenzelm@10277:     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
wenzelm@10277:   and [simp] = Rep_multiset_inject [symmetric]
wenzelm@10249: 
wenzelm@19086: definition
wenzelm@21404:   Mempty :: "'a multiset"  ("{#}") where
wenzelm@19086:   "{#} = Abs_multiset (\<lambda>a. 0)"
wenzelm@10249: 
wenzelm@21404: definition
nipkow@25507:   single :: "'a => 'a multiset" where
nipkow@25507:   "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
wenzelm@10249: 
wenzelm@21404: definition
wenzelm@21404:   count :: "'a multiset => 'a => nat" where
wenzelm@19086:   "count = Rep_multiset"
wenzelm@10249: 
wenzelm@21404: definition
wenzelm@21404:   MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
wenzelm@19086:   "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
wenzelm@19086: 
wenzelm@19363: abbreviation
wenzelm@21404:   Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
nipkow@25162:   "a :# M == count M a > 0"
wenzelm@10249: 
wenzelm@10249: syntax
wenzelm@10249:   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
wenzelm@10249: translations
wenzelm@20770:   "{#x:M. P#}" == "CONST MCollect M (\<lambda>x. P)"
wenzelm@10249: 
wenzelm@19086: definition
wenzelm@21404:   set_of :: "'a multiset => 'a set" where
wenzelm@19086:   "set_of M = {x. x :# M}"
wenzelm@10249: 
haftmann@21417: instance multiset :: (type) "{plus, minus, zero, size}" 
nipkow@11464:   union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
nipkow@11464:   diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
wenzelm@11701:   Zero_multiset_def [simp]: "0 == {#}"
haftmann@21417:   size_def: "size M == setsum (count M) (set_of M)" ..
wenzelm@10249: 
wenzelm@19086: definition
wenzelm@21404:   multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
wenzelm@19086:   "multiset_inter A B = A - (A - B)"
kleing@15869: 
nipkow@25507: syntax -- "Multiset Enumeration"
nipkow@25507:   "@multiset" :: "args => 'a multiset"    ("{#(_)#}")
nipkow@25507: 
nipkow@25507: translations
nipkow@25507:   "{#x, xs#}" == "{#x#} + {#xs#}"
nipkow@25507:   "{#x#}" == "CONST single x"
nipkow@25507: 
wenzelm@10249: 
wenzelm@10249: text {*
wenzelm@10249:  \medskip Preservation of the representing set @{term multiset}.
wenzelm@10249: *}
wenzelm@10249: 
nipkow@11464: lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
wenzelm@17161:   by (simp add: multiset_def)
wenzelm@10249: 
wenzelm@11701: lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
wenzelm@17161:   by (simp add: multiset_def)
wenzelm@10249: 
wenzelm@10249: lemma union_preserves_multiset [simp]:
nipkow@11464:     "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
wenzelm@17161:   apply (simp add: multiset_def)
wenzelm@17161:   apply (drule (1) finite_UnI)
wenzelm@10249:   apply (simp del: finite_Un add: Un_def)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: lemma diff_preserves_multiset [simp]:
nipkow@11464:     "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
wenzelm@17161:   apply (simp add: multiset_def)
wenzelm@10249:   apply (rule finite_subset)
wenzelm@17161:    apply auto
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsection {* Algebraic properties of multisets *}
wenzelm@10249: 
wenzelm@10249: subsubsection {* Union *}
wenzelm@10249: 
wenzelm@17161: lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
wenzelm@17161:   by (simp add: union_def Mempty_def)
wenzelm@10249: 
wenzelm@17161: lemma union_commute: "M + N = N + (M::'a multiset)"
wenzelm@17161:   by (simp add: union_def add_ac)
wenzelm@17161: 
wenzelm@17161: lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
wenzelm@17161:   by (simp add: union_def add_ac)
wenzelm@10249: 
wenzelm@17161: lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
wenzelm@17161: proof -
wenzelm@17161:   have "M + (N + K) = (N + K) + M"
wenzelm@17161:     by (rule union_commute)
wenzelm@17161:   also have "\<dots> = N + (K + M)"
wenzelm@17161:     by (rule union_assoc)
wenzelm@17161:   also have "K + M = M + K"
wenzelm@17161:     by (rule union_commute)
wenzelm@17161:   finally show ?thesis .
wenzelm@17161: qed
wenzelm@10249: 
wenzelm@17161: lemmas union_ac = union_assoc union_commute union_lcomm
wenzelm@10249: 
obua@14738: instance multiset :: (type) comm_monoid_add
wenzelm@17200: proof
obua@14722:   fix a b c :: "'a multiset"
obua@14722:   show "(a + b) + c = a + (b + c)" by (rule union_assoc)
obua@14722:   show "a + b = b + a" by (rule union_commute)
obua@14722:   show "0 + a = a" by simp
obua@14722: qed
wenzelm@10277: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Difference *}
wenzelm@10249: 
wenzelm@17161: lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
wenzelm@17161:   by (simp add: Mempty_def diff_def)
wenzelm@10249: 
wenzelm@17161: lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
wenzelm@17161:   by (simp add: union_def diff_def)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Count of elements *}
wenzelm@10249: 
wenzelm@17161: lemma count_empty [simp]: "count {#} a = 0"
wenzelm@17161:   by (simp add: count_def Mempty_def)
wenzelm@10249: 
wenzelm@17161: lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
wenzelm@17161:   by (simp add: count_def single_def)
wenzelm@10249: 
wenzelm@17161: lemma count_union [simp]: "count (M + N) a = count M a + count N a"
wenzelm@17161:   by (simp add: count_def union_def)
wenzelm@10249: 
wenzelm@17161: lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
wenzelm@17161:   by (simp add: count_def diff_def)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Set of elements *}
wenzelm@10249: 
wenzelm@17161: lemma set_of_empty [simp]: "set_of {#} = {}"
wenzelm@17161:   by (simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_single [simp]: "set_of {#b#} = {b}"
wenzelm@17161:   by (simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
wenzelm@17161:   by (auto simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
wenzelm@17161:   by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
wenzelm@17161:   by (auto simp add: set_of_def)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Size *}
wenzelm@10249: 
wenzelm@17161: lemma size_empty [simp]: "size {#} = 0"
wenzelm@17161:   by (simp add: size_def)
wenzelm@10249: 
wenzelm@17161: lemma size_single [simp]: "size {#b#} = 1"
wenzelm@17161:   by (simp add: size_def)
wenzelm@10249: 
wenzelm@17161: lemma finite_set_of [iff]: "finite (set_of M)"
wenzelm@17161:   using Rep_multiset [of M]
wenzelm@17161:   by (simp add: multiset_def set_of_def count_def)
wenzelm@10249: 
wenzelm@17161: lemma setsum_count_Int:
nipkow@11464:     "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
wenzelm@18258:   apply (induct rule: finite_induct)
wenzelm@17161:    apply simp
wenzelm@10249:   apply (simp add: Int_insert_left set_of_def)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
wenzelm@10249:   apply (unfold size_def)
nipkow@11464:   apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
wenzelm@10249:    prefer 2
paulson@15072:    apply (rule ext, simp)
nipkow@15402:   apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
wenzelm@10249:   apply (subst Int_commute)
wenzelm@10249:   apply (simp (no_asm_simp) add: setsum_count_Int)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
paulson@15072:   apply (unfold size_def Mempty_def count_def, auto)
wenzelm@10249:   apply (simp add: set_of_def count_def expand_fun_eq)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
wenzelm@10249:   apply (unfold size_def)
paulson@15072:   apply (drule setsum_SucD, auto)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Equality of multisets *}
wenzelm@10249: 
wenzelm@17161: lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
wenzelm@17161:   by (simp add: count_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
wenzelm@17161:   by (simp add: single_def Mempty_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
wenzelm@17161:   by (auto simp add: single_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
wenzelm@17161:   by (auto simp add: union_def Mempty_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
wenzelm@17161:   by (auto simp add: union_def Mempty_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
wenzelm@17161:   by (simp add: union_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
wenzelm@17161:   by (simp add: union_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_is_single:
nipkow@11464:     "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
paulson@15072:   apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq)
wenzelm@10249:   apply blast
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma single_is_union:
paulson@15072:      "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
wenzelm@10249:   apply (unfold Mempty_def single_def union_def)
nipkow@11464:   apply (simp add: add_is_1 one_is_add expand_fun_eq)
wenzelm@10249:   apply (blast dest: sym)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma add_eq_conv_diff:
wenzelm@10249:   "(M + {#a#} = N + {#b#}) =
paulson@15072:    (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
wenzelm@24035:   using [[simproc del: neq]]
wenzelm@10249:   apply (unfold single_def union_def diff_def)
wenzelm@10249:   apply (simp (no_asm) add: expand_fun_eq)
paulson@15072:   apply (rule conjI, force, safe, simp_all)
berghofe@13601:   apply (simp add: eq_sym_conv)
wenzelm@10249:   done
wenzelm@10249: 
kleing@15869: declare Rep_multiset_inject [symmetric, simp del]
kleing@15869: 
nipkow@23611: instance multiset :: (type) cancel_ab_semigroup_add
nipkow@23611: proof
nipkow@23611:   fix a b c :: "'a multiset"
nipkow@23611:   show "a + b = a + c \<Longrightarrow> b = c" by simp
nipkow@23611: qed
kleing@15869: 
kleing@15869: subsubsection {* Intersection *}
kleing@15869: 
kleing@15869: lemma multiset_inter_count:
wenzelm@17161:     "count (A #\<inter> B) x = min (count A x) (count B x)"
wenzelm@17161:   by (simp add: multiset_inter_def min_def)
kleing@15869: 
kleing@15869: lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
wenzelm@17200:   by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214:     min_max.inf_commute)
kleing@15869: 
kleing@15869: lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
wenzelm@17200:   by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214:     min_max.inf_assoc)
kleing@15869: 
kleing@15869: lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
kleing@15869:   by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
kleing@15869: 
wenzelm@17161: lemmas multiset_inter_ac =
wenzelm@17161:   multiset_inter_commute
wenzelm@17161:   multiset_inter_assoc
wenzelm@17161:   multiset_inter_left_commute
kleing@15869: 
kleing@15869: lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
wenzelm@17200:   apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
wenzelm@17161:     split: split_if_asm)
kleing@15869:   apply clarsimp
wenzelm@17161:   apply (erule_tac x = a in allE)
kleing@15869:   apply auto
kleing@15869:   done
kleing@15869: 
wenzelm@10249: 
wenzelm@10249: subsection {* Induction over multisets *}
wenzelm@10249: 
wenzelm@10249: lemma setsum_decr:
wenzelm@11701:   "finite F ==> (0::nat) < f a ==>
paulson@15072:     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
wenzelm@18258:   apply (induct rule: finite_induct)
wenzelm@18258:    apply auto
paulson@15072:   apply (drule_tac a = a in mk_disjoint_insert, auto)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10313: lemma rep_multiset_induct_aux:
wenzelm@18730:   assumes 1: "P (\<lambda>a. (0::nat))"
wenzelm@18730:     and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow@25134:   shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
wenzelm@18730:   apply (unfold multiset_def)
wenzelm@18730:   apply (induct_tac n, simp, clarify)
wenzelm@18730:    apply (subgoal_tac "f = (\<lambda>a.0)")
wenzelm@18730:     apply simp
wenzelm@18730:     apply (rule 1)
wenzelm@18730:    apply (rule ext, force, clarify)
wenzelm@18730:   apply (frule setsum_SucD, clarify)
wenzelm@18730:   apply (rename_tac a)
nipkow@25162:   apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
wenzelm@18730:    prefer 2
wenzelm@18730:    apply (rule finite_subset)
wenzelm@18730:     prefer 2
wenzelm@18730:     apply assumption
wenzelm@18730:    apply simp
wenzelm@18730:    apply blast
wenzelm@18730:   apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
wenzelm@18730:    prefer 2
wenzelm@18730:    apply (rule ext)
wenzelm@18730:    apply (simp (no_asm_simp))
wenzelm@18730:    apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
wenzelm@18730:   apply (erule allE, erule impE, erule_tac [2] mp, blast)
wenzelm@18730:   apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow@25134:   apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
wenzelm@18730:    prefer 2
wenzelm@18730:    apply blast
nipkow@25134:   apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
wenzelm@18730:    prefer 2
wenzelm@18730:    apply blast
wenzelm@18730:   apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
wenzelm@18730:   done
wenzelm@10249: 
wenzelm@10313: theorem rep_multiset_induct:
nipkow@11464:   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701:     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
wenzelm@17161:   using rep_multiset_induct_aux by blast
wenzelm@10249: 
wenzelm@18258: theorem multiset_induct [case_names empty add, induct type: multiset]:
wenzelm@18258:   assumes empty: "P {#}"
wenzelm@18258:     and add: "!!M x. P M ==> P (M + {#x#})"
wenzelm@17161:   shows "P M"
wenzelm@10249: proof -
wenzelm@10249:   note defns = union_def single_def Mempty_def
wenzelm@10249:   show ?thesis
wenzelm@10249:     apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313:     apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@18258:      apply (rule empty [unfolded defns])
paulson@15072:     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249:      prefer 2
wenzelm@10249:      apply (simp add: expand_fun_eq)
wenzelm@10249:     apply (erule ssubst)
wenzelm@17200:     apply (erule Abs_multiset_inverse [THEN subst])
wenzelm@18258:     apply (erule add [unfolded defns, simplified])
wenzelm@10249:     done
wenzelm@10249: qed
wenzelm@10249: 
wenzelm@10249: lemma MCollect_preserves_multiset:
nipkow@11464:     "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
wenzelm@10249:   apply (simp add: multiset_def)
paulson@15072:   apply (rule finite_subset, auto)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma count_MCollect [simp]:
wenzelm@10249:     "count {# x:M. P x #} a = (if P a then count M a else 0)"
paulson@15072:   by (simp add: count_def MCollect_def MCollect_preserves_multiset)
wenzelm@10249: 
wenzelm@17161: lemma set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
wenzelm@17161:   by (auto simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
wenzelm@17161:   by (subst multiset_eq_conv_count_eq, auto)
wenzelm@10249: 
wenzelm@17161: lemma add_eq_conv_ex:
wenzelm@17161:   "(M + {#a#} = N + {#b#}) =
wenzelm@17161:     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
paulson@15072:   by (auto simp add: add_eq_conv_diff)
wenzelm@10249: 
kleing@15869: declare multiset_typedef [simp del]
wenzelm@10249: 
wenzelm@17161: 
wenzelm@10249: subsection {* Multiset orderings *}
wenzelm@10249: 
wenzelm@10249: subsubsection {* Well-foundedness *}
wenzelm@10249: 
wenzelm@19086: definition
berghofe@23751:   mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
wenzelm@19086:   "mult1 r =
berghofe@23751:     {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
berghofe@23751:       (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249: 
wenzelm@21404: definition
berghofe@23751:   mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
berghofe@23751:   "mult r = (mult1 r)\<^sup>+"
wenzelm@10249: 
berghofe@23751: lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
wenzelm@10277:   by (simp add: mult1_def)
wenzelm@10249: 
berghofe@23751: lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
berghofe@23751:     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
berghofe@23751:     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@19582:   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
wenzelm@10249: proof (unfold mult1_def)
berghofe@23751:   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (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"
berghofe@23751:   let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249: 
berghofe@23751:   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
wenzelm@18258:   then have "\<exists>a' M0' K.
nipkow@11464:       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
wenzelm@18258:   then show "?case1 \<or> ?case2"
wenzelm@10249:   proof (elim exE conjE)
wenzelm@10249:     fix a' M0' K
wenzelm@10249:     assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249:     assume "M0 + {#a#} = M0' + {#a'#}"
wenzelm@18258:     then have "M0 = M0' \<and> a = a' \<or>
nipkow@11464:         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249:       by (simp only: add_eq_conv_ex)
wenzelm@18258:     then show ?thesis
wenzelm@10249:     proof (elim disjE conjE exE)
wenzelm@10249:       assume "M0 = M0'" "a = a'"
nipkow@11464:       with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@18258:       then have ?case2 .. then show ?thesis ..
wenzelm@10249:     next
wenzelm@10249:       fix K'
wenzelm@10249:       assume "M0' = K' + {#a#}"
wenzelm@10249:       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249: 
wenzelm@10249:       assume "M0 = K' + {#a'#}"
wenzelm@10249:       with r have "?R (K' + K) M0" by blast
wenzelm@18258:       with n have ?case1 by simp then show ?thesis ..
wenzelm@10249:     qed
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
berghofe@23751: lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249: proof
wenzelm@10249:   let ?R = "mult1 r"
wenzelm@10249:   let ?W = "acc ?R"
wenzelm@10249:   {
wenzelm@10249:     fix M M0 a
berghofe@23751:     assume M0: "M0 \<in> ?W"
berghofe@23751:       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751:       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> 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"
berghofe@23751:       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
berghofe@23751:           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249:         by (rule less_add)
berghofe@23751:       then show "N \<in> ?W"
wenzelm@10249:       proof (elim exE disjE conjE)
berghofe@23751:         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
berghofe@23751:         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
berghofe@23751:         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
berghofe@23751:         then show "N \<in> ?W" by (simp only: N)
wenzelm@10249:       next
wenzelm@10249:         fix K
wenzelm@10249:         assume N: "N = M0 + K"
berghofe@23751:         assume "\<forall>b. b :# K --> (b, a) \<in> r"
berghofe@23751:         then 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" ..
berghofe@23751:           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249:         qed
berghofe@23751:         then show "N \<in> ?W" by (simp only: N)
wenzelm@10249:       qed
wenzelm@10249:     qed
wenzelm@10249:   } note tedious_reasoning = this
wenzelm@10249: 
berghofe@23751:   assume wf: "wf r"
wenzelm@10249:   fix M
berghofe@23751:   show "M \<in> ?W"
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"
berghofe@23751:     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:     proof induct
wenzelm@10249:       fix a
berghofe@23751:       assume r: "!!b. (b, a) \<in> r ==> (\<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
berghofe@23751:     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
berghofe@23751: theorem wf_mult1: "wf r ==> wf (mult1 r)"
wenzelm@23373:   by (rule acc_wfI) (rule all_accessible)
wenzelm@10249: 
berghofe@23751: theorem wf_mult: "wf r ==> wf (mult r)"
berghofe@23751:   unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Closure-free presentation *}
wenzelm@10249: 
wenzelm@10249: (*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249: 
wenzelm@10249: lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
wenzelm@23373:   by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249: 
wenzelm@10249: text {* One direction. *}
wenzelm@10249: 
wenzelm@10249: lemma mult_implies_one_step:
berghofe@23751:   "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464:     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
berghofe@23751:     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
wenzelm@10249:   apply (unfold mult_def mult1_def set_of_def)
berghofe@23751:   apply (erule converse_trancl_induct, clarify)
paulson@15072:    apply (rule_tac x = M0 in exI, simp, clarify)
berghofe@23751:   apply (case_tac "a :# K")
wenzelm@10249:    apply (rule_tac x = I in exI)
wenzelm@10249:    apply (simp (no_asm))
berghofe@23751:    apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
wenzelm@10249:    apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@11464:    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
wenzelm@10249:    apply (simp add: diff_union_single_conv)
wenzelm@10249:    apply (simp (no_asm_use) add: trans_def)
wenzelm@10249:    apply blast
wenzelm@10249:   apply (subgoal_tac "a :# I")
wenzelm@10249:    apply (rule_tac x = "I - {#a#}" in exI)
wenzelm@10249:    apply (rule_tac x = "J + {#a#}" in exI)
wenzelm@10249:    apply (rule_tac x = "K + Ka" in exI)
wenzelm@10249:    apply (rule conjI)
wenzelm@10249:     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249:    apply (rule conjI)
paulson@15072:     apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
wenzelm@10249:     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249:    apply (simp (no_asm_use) add: trans_def)
wenzelm@10249:    apply blast
wenzelm@10277:   apply (subgoal_tac "a :# (M0 + {#a#})")
wenzelm@10249:    apply simp
wenzelm@10249:   apply (simp (no_asm))
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
wenzelm@23373:   by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249: 
nipkow@11464: lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
wenzelm@10249:   apply (erule size_eq_Suc_imp_elem [THEN exE])
paulson@15072:   apply (drule elem_imp_eq_diff_union, auto)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: lemma one_step_implies_mult_aux:
berghofe@23751:   "trans r ==>
berghofe@23751:     \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
berghofe@23751:       --> (I + K, I + J) \<in> mult r"
paulson@15072:   apply (induct_tac n, auto)
paulson@15072:   apply (frule size_eq_Suc_imp_eq_union, clarify)
paulson@15072:   apply (rename_tac "J'", simp)
paulson@15072:   apply (erule notE, auto)
wenzelm@10249:   apply (case_tac "J' = {#}")
wenzelm@10249:    apply (simp add: mult_def)
berghofe@23751:    apply (rule r_into_trancl)
paulson@15072:    apply (simp add: mult1_def set_of_def, blast)
nipkow@11464:   txt {* Now we know @{term "J' \<noteq> {#}"}. *}
berghofe@23751:   apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@11464:   apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
wenzelm@10249:   apply (erule ssubst)
paulson@15072:   apply (simp add: Ball_def, auto)
wenzelm@10249:   apply (subgoal_tac
berghofe@23751:     "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
berghofe@23751:       (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
wenzelm@10249:    prefer 2
wenzelm@10249:    apply force
wenzelm@10249:   apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
berghofe@23751:   apply (erule trancl_trans)
berghofe@23751:   apply (rule r_into_trancl)
wenzelm@10249:   apply (simp add: mult1_def set_of_def)
wenzelm@10249:   apply (rule_tac x = a in exI)
wenzelm@10249:   apply (rule_tac x = "I + J'" in exI)
wenzelm@10249:   apply (simp add: union_ac)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma one_step_implies_mult:
berghofe@23751:   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
berghofe@23751:     ==> (I + K, I + J) \<in> mult r"
wenzelm@23373:   using one_step_implies_mult_aux by blast
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Partial-order properties *}
wenzelm@10249: 
wenzelm@12338: instance multiset :: (type) ord ..
wenzelm@10249: 
wenzelm@10249: defs (overloaded)
berghofe@23751:   less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
nipkow@11464:   le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249: 
berghofe@23751: lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
wenzelm@18730:   unfolding trans_def by (blast intro: order_less_trans)
wenzelm@10249: 
wenzelm@10249: text {*
wenzelm@10249:  \medskip Irreflexivity.
wenzelm@10249: *}
wenzelm@10249: 
wenzelm@10249: lemma mult_irrefl_aux:
wenzelm@18258:     "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
wenzelm@23373:   by (induct rule: finite_induct) (auto intro: order_less_trans)
wenzelm@10249: 
wenzelm@17161: lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
paulson@15072:   apply (unfold less_multiset_def, auto)
paulson@15072:   apply (drule trans_base_order [THEN mult_implies_one_step], auto)
wenzelm@10249:   apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
wenzelm@10249:   apply (simp add: set_of_eq_empty_iff)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
wenzelm@23373:   using insert mult_less_not_refl by fast
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: text {* Transitivity. *}
wenzelm@10249: 
wenzelm@10249: theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
berghofe@23751:   unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@10249: 
wenzelm@10249: text {* Asymmetry. *}
wenzelm@10249: 
nipkow@11464: theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
wenzelm@10249:   apply auto
wenzelm@10249:   apply (rule mult_less_not_refl [THEN notE])
paulson@15072:   apply (erule mult_less_trans, assumption)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: theorem mult_less_asym:
nipkow@11464:     "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
paulson@15072:   by (insert mult_less_not_sym, blast)
wenzelm@10249: 
wenzelm@10249: theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
wenzelm@18730:   unfolding le_multiset_def by auto
wenzelm@10249: 
wenzelm@10249: text {* Anti-symmetry. *}
wenzelm@10249: 
wenzelm@10249: theorem mult_le_antisym:
wenzelm@10249:     "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
wenzelm@18730:   unfolding le_multiset_def by (blast dest: mult_less_not_sym)
wenzelm@10249: 
wenzelm@10249: text {* Transitivity. *}
wenzelm@10249: 
wenzelm@10249: theorem mult_le_trans:
wenzelm@10249:     "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
wenzelm@18730:   unfolding le_multiset_def by (blast intro: mult_less_trans)
wenzelm@10249: 
wenzelm@11655: theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
wenzelm@18730:   unfolding le_multiset_def by auto
wenzelm@10249: 
wenzelm@10277: text {* Partial order. *}
wenzelm@10277: 
wenzelm@10277: instance multiset :: (order) order
wenzelm@10277:   apply intro_classes
berghofe@23751:   apply (rule mult_less_le)
berghofe@23751:   apply (rule mult_le_refl)
berghofe@23751:   apply (erule mult_le_trans, assumption)
berghofe@23751:   apply (erule mult_le_antisym, assumption)
wenzelm@10277:   done
wenzelm@10277: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Monotonicity of multiset union *}
wenzelm@10249: 
wenzelm@17161: lemma mult1_union:
berghofe@23751:     "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
paulson@15072:   apply (unfold mult1_def, auto)
wenzelm@10249:   apply (rule_tac x = a in exI)
wenzelm@10249:   apply (rule_tac x = "C + M0" in exI)
wenzelm@10249:   apply (simp add: union_assoc)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
wenzelm@10249:   apply (unfold less_multiset_def mult_def)
berghofe@23751:   apply (erule trancl_induct)
berghofe@23751:    apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
berghofe@23751:   apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@10249: lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
wenzelm@10249:   apply (subst union_commute [of B C])
wenzelm@10249:   apply (subst union_commute [of D C])
wenzelm@10249:   apply (erule union_less_mono2)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma union_less_mono:
wenzelm@10249:     "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
wenzelm@10249:   apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma union_le_mono:
wenzelm@10249:     "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
wenzelm@18730:   unfolding le_multiset_def
wenzelm@18730:   by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249: 
wenzelm@17161: lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
wenzelm@10249:   apply (unfold le_multiset_def less_multiset_def)
wenzelm@10249:   apply (case_tac "M = {#}")
wenzelm@10249:    prefer 2
berghofe@23751:    apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
wenzelm@10249:     prefer 2
wenzelm@10249:     apply (rule one_step_implies_mult)
berghofe@23751:       apply (simp only: trans_def, auto)
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@17161: lemma union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072: proof -
wenzelm@17200:   have "A + {#} <= A + B" by (blast intro: union_le_mono)
wenzelm@18258:   then show ?thesis by simp
paulson@15072: qed
paulson@15072: 
wenzelm@17161: lemma union_upper2: "B <= A + (B::'a::order multiset)"
wenzelm@18258:   by (subst union_commute) (rule union_upper1)
paulson@15072: 
nipkow@23611: instance multiset :: (order) pordered_ab_semigroup_add
nipkow@23611: apply intro_classes
nipkow@23611: apply(erule union_le_mono[OF mult_le_refl])
nipkow@23611: done
paulson@15072: 
wenzelm@17200: subsection {* Link with lists *}
paulson@15072: 
wenzelm@17200: consts
paulson@15072:   multiset_of :: "'a list \<Rightarrow> 'a multiset"
paulson@15072: primrec
paulson@15072:   "multiset_of [] = {#}"
paulson@15072:   "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072: 
paulson@15072: lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
wenzelm@18258:   by (induct x) auto
paulson@15072: 
paulson@15072: lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
wenzelm@18258:   by (induct x) auto
paulson@15072: 
paulson@15072: lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
wenzelm@18258:   by (induct x) auto
kleing@15867: 
kleing@15867: lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
kleing@15867:   by (induct xs) auto
paulson@15072: 
wenzelm@18258: lemma multiset_of_append [simp]:
wenzelm@18258:     "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
wenzelm@20503:   by (induct xs arbitrary: ys) (auto simp: union_ac)
wenzelm@18730: 
paulson@15072: lemma surj_multiset_of: "surj multiset_of"
wenzelm@17200:   apply (unfold surj_def, rule allI)
wenzelm@17200:   apply (rule_tac M=y in multiset_induct, auto)
wenzelm@17200:   apply (rule_tac x = "x # xa" in exI, auto)
wenzelm@10249:   done
wenzelm@10249: 
nipkow@25162: lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
wenzelm@18258:   by (induct x) auto
paulson@15072: 
wenzelm@17200: lemma distinct_count_atmost_1:
paulson@15072:    "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
wenzelm@18258:    apply (induct x, simp, rule iffI, simp_all)
wenzelm@17200:    apply (rule conjI)
wenzelm@17200:    apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
paulson@15072:    apply (erule_tac x=a in allE, simp, clarify)
wenzelm@17200:    apply (erule_tac x=aa in allE, simp)
paulson@15072:    done
paulson@15072: 
wenzelm@17200: lemma multiset_of_eq_setD:
kleing@15867:   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
kleing@15867:   by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
kleing@15867: 
wenzelm@17200: lemma set_eq_iff_multiset_of_eq_distinct:
wenzelm@17200:   "\<lbrakk>distinct x; distinct y\<rbrakk>
paulson@15072:    \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
wenzelm@17200:   by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
paulson@15072: 
wenzelm@17200: lemma set_eq_iff_multiset_of_remdups_eq:
paulson@15072:    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
wenzelm@17200:   apply (rule iffI)
wenzelm@17200:   apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
wenzelm@17200:   apply (drule distinct_remdups[THEN distinct_remdups
wenzelm@17200:                       [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]])
paulson@15072:   apply simp
wenzelm@10249:   done
wenzelm@10249: 
wenzelm@18258: lemma multiset_of_compl_union [simp]:
nipkow@23281:     "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
kleing@15630:   by (induct xs) (auto simp: union_ac)
paulson@15072: 
wenzelm@17200: lemma count_filter:
nipkow@23281:     "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
wenzelm@18258:   by (induct xs) auto
kleing@15867: 
kleing@15867: 
paulson@15072: subsection {* Pointwise ordering induced by count *}
paulson@15072: 
wenzelm@19086: definition
nipkow@23611: mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
nipkow@23611: "(A \<le># B) = (\<forall>a. count A a \<le> count B a)"
nipkow@23611: definition
nipkow@23611: mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
nipkow@23611: "(A <# B) = (A \<le># B \<and> A \<noteq> B)"
paulson@15072: 
nipkow@23611: lemma mset_le_refl[simp]: "A \<le># A"
wenzelm@18730:   unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_trans: "\<lbrakk> A \<le># B; B \<le># C \<rbrakk> \<Longrightarrow> A \<le># C"
wenzelm@18730:   unfolding mset_le_def by (fast intro: order_trans)
paulson@15072: 
nipkow@23611: lemma mset_le_antisym: "\<lbrakk> A \<le># B; B \<le># A \<rbrakk> \<Longrightarrow> A = B"
wenzelm@17200:   apply (unfold mset_le_def)
wenzelm@17200:   apply (rule multiset_eq_conv_count_eq[THEN iffD2])
paulson@15072:   apply (blast intro: order_antisym)
paulson@15072:   done
paulson@15072: 
wenzelm@17200: lemma mset_le_exists_conv:
nipkow@23611:   "(A \<le># B) = (\<exists>C. B = A + C)"
nipkow@23611:   apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
paulson@15072:   apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
paulson@15072:   done
paulson@15072: 
nipkow@23611: lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
wenzelm@18730:   unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
wenzelm@18730:   unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
wenzelm@17200:   apply (unfold mset_le_def)
wenzelm@17200:   apply auto
paulson@15072:   apply (erule_tac x=a in allE)+
paulson@15072:   apply auto
paulson@15072:   done
paulson@15072: 
nipkow@23611: lemma mset_le_add_left[simp]: "A \<le># A + B"
wenzelm@18730:   unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_add_right[simp]: "B \<le># A + B"
wenzelm@18730:   unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
nipkow@23611: apply (induct xs)
nipkow@23611:  apply auto
nipkow@23611: apply (rule mset_le_trans)
nipkow@23611:  apply auto
nipkow@23611: done
nipkow@23611: 
haftmann@25208: interpretation mset_order:
haftmann@25208:   order ["op \<le>#" "op <#"]
haftmann@25208:   by (auto intro: order.intro mset_le_refl mset_le_antisym
haftmann@25208:     mset_le_trans simp: mset_less_def)
nipkow@23611: 
nipkow@23611: interpretation mset_order_cancel_semigroup:
haftmann@25208:   pordered_cancel_ab_semigroup_add ["op \<le>#" "op <#" "op +"]
haftmann@25208:   by unfold_locales (erule mset_le_mono_add [OF mset_le_refl])
nipkow@23611: 
nipkow@23611: interpretation mset_order_semigroup_cancel:
haftmann@25208:   pordered_ab_semigroup_add_imp_le ["op \<le>#" "op <#" "op +"]
haftmann@25208:   by (unfold_locales) simp
paulson@15072: 
wenzelm@10249: end




