src/HOL/Library/Multiset.thy
author wenzelm
Wed Aug 31 15:46:37 2005 +0200 (2005-08-31)
changeset 17200 3a4d03d1a31b
parent 17161 57c69627d71a
child 17778 93d7e524417a
permissions -rw-r--r--
tuned presentation;
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(*  Title:      HOL/Library/Multiset.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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*)
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header {* Multisets *}
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theory Multiset
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imports Accessible_Part
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begin
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subsection {* The type of multisets *}
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typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
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proof
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  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
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qed
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lemmas multiset_typedef [simp] =
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    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
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  and [simp] = Rep_multiset_inject [symmetric]
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constdefs
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  Mempty :: "'a multiset"    ("{#}")
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  "{#} == Abs_multiset (\<lambda>a. 0)"
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  single :: "'a => 'a multiset"    ("{#_#}")
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  "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
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  count :: "'a multiset => 'a => nat"
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  "count == Rep_multiset"
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  MCollect :: "'a multiset => ('a => bool) => 'a multiset"
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  "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
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syntax
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  "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
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  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
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translations
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  "a :# M" == "0 < count M a"
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  "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
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constdefs
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  set_of :: "'a multiset => 'a set"
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  "set_of M == {x. x :# M}"
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instance multiset :: (type) "{plus, minus, zero}" ..
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defs (overloaded)
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  union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
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  diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
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  Zero_multiset_def [simp]: "0 == {#}"
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  size_def: "size M == setsum (count M) (set_of M)"
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constdefs
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 multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70)
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 "multiset_inter A B \<equiv> A - (A - B)"
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text {*
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 \medskip Preservation of the representing set @{term multiset}.
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*}
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lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma union_preserves_multiset [simp]:
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    "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
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  apply (simp add: multiset_def)
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  apply (drule (1) finite_UnI)
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  apply (simp del: finite_Un add: Un_def)
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  done
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lemma diff_preserves_multiset [simp]:
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    "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
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  apply (simp add: multiset_def)
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  apply (rule finite_subset)
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   apply auto
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  done
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subsection {* Algebraic properties of multisets *}
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subsubsection {* Union *}
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lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
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  by (simp add: union_def Mempty_def)
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lemma union_commute: "M + N = N + (M::'a multiset)"
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  by (simp add: union_def add_ac)
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lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
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  by (simp add: union_def add_ac)
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lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
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proof -
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  have "M + (N + K) = (N + K) + M"
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    by (rule union_commute)
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  also have "\<dots> = N + (K + M)"
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    by (rule union_assoc)
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  also have "K + M = M + K"
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    by (rule union_commute)
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  finally show ?thesis .
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qed
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lemmas union_ac = union_assoc union_commute union_lcomm
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instance multiset :: (type) comm_monoid_add
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proof
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  fix a b c :: "'a multiset"
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  show "(a + b) + c = a + (b + c)" by (rule union_assoc)
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  show "a + b = b + a" by (rule union_commute)
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  show "0 + a = a" by simp
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qed
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subsubsection {* Difference *}
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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  by (simp add: Mempty_def diff_def)
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lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
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  by (simp add: union_def diff_def)
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subsubsection {* Count of elements *}
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lemma count_empty [simp]: "count {#} a = 0"
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  by (simp add: count_def Mempty_def)
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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  by (simp add: count_def single_def)
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lemma count_union [simp]: "count (M + N) a = count M a + count N a"
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  by (simp add: count_def union_def)
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
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  by (simp add: count_def diff_def)
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subsubsection {* Set of elements *}
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lemma set_of_empty [simp]: "set_of {#} = {}"
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  by (simp add: set_of_def)
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lemma set_of_single [simp]: "set_of {#b#} = {b}"
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  by (simp add: set_of_def)
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lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
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  by (auto simp add: set_of_def)
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lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
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  by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
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lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
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  by (auto simp add: set_of_def)
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subsubsection {* Size *}
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lemma size_empty [simp]: "size {#} = 0"
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  by (simp add: size_def)
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lemma size_single [simp]: "size {#b#} = 1"
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  by (simp add: size_def)
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lemma finite_set_of [iff]: "finite (set_of M)"
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  using Rep_multiset [of M]
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  by (simp add: multiset_def set_of_def count_def)
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lemma setsum_count_Int:
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    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
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  apply (erule finite_induct)
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   apply simp
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  apply (simp add: Int_insert_left set_of_def)
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  done
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lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
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  apply (unfold size_def)
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  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
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   prefer 2
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   apply (rule ext, simp)
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  apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
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  apply (subst Int_commute)
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  apply (simp (no_asm_simp) add: setsum_count_Int)
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  done
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lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
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  apply (unfold size_def Mempty_def count_def, auto)
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  apply (simp add: set_of_def count_def expand_fun_eq)
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  done
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lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
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  apply (unfold size_def)
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  apply (drule setsum_SucD, auto)
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  done
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subsubsection {* Equality of multisets *}
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lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
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  by (simp add: count_def expand_fun_eq)
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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  by (simp add: single_def Mempty_def expand_fun_eq)
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lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
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  by (auto simp add: single_def expand_fun_eq)
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lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
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  by (auto simp add: union_def Mempty_def expand_fun_eq)
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lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
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  by (auto simp add: union_def Mempty_def expand_fun_eq)
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lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
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  by (simp add: union_def expand_fun_eq)
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lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
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  by (simp add: union_def expand_fun_eq)
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lemma union_is_single:
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    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
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  apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq)
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  apply blast
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  done
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lemma single_is_union:
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     "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
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  apply (unfold Mempty_def single_def union_def)
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  apply (simp add: add_is_1 one_is_add expand_fun_eq)
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  apply (blast dest: sym)
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  done
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lemma add_eq_conv_diff:
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  "(M + {#a#} = N + {#b#}) =
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   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
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  apply (unfold single_def union_def diff_def)
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  apply (simp (no_asm) add: expand_fun_eq)
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  apply (rule conjI, force, safe, simp_all)
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  apply (simp add: eq_sym_conv)
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  done
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declare Rep_multiset_inject [symmetric, simp del]
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subsubsection {* Intersection *}
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lemma multiset_inter_count:
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    "count (A #\<inter> B) x = min (count A x) (count B x)"
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  by (simp add: multiset_inter_def min_def)
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lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
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  by (simp add: multiset_eq_conv_count_eq multiset_inter_count
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    min_max.below_inf.inf_commute)
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lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
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  by (simp add: multiset_eq_conv_count_eq multiset_inter_count
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    min_max.below_inf.inf_assoc)
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lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
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  by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
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lemmas multiset_inter_ac =
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  multiset_inter_commute
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  multiset_inter_assoc
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  multiset_inter_left_commute
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lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
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  apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
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    split: split_if_asm)
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  apply clarsimp
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  apply (erule_tac x = a in allE)
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  apply auto
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  done
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subsection {* Induction over multisets *}
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lemma setsum_decr:
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  "finite F ==> (0::nat) < f a ==>
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    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
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  apply (erule finite_induct, auto)
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  apply (drule_tac a = a in mk_disjoint_insert, auto)
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  done
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lemma rep_multiset_induct_aux:
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  assumes "P (\<lambda>a. (0::nat))"
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    and "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
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  shows "\<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
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proof -
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  note premises = prems [unfolded multiset_def]
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  show ?thesis
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    apply (unfold multiset_def)
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    apply (induct_tac n, simp, clarify)
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     apply (subgoal_tac "f = (\<lambda>a.0)")
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      apply simp
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      apply (rule premises)
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     apply (rule ext, force, clarify)
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    apply (frule setsum_SucD, clarify)
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    apply (rename_tac a)
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    apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
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     prefer 2
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     apply (rule finite_subset)
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      prefer 2
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      apply assumption
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     apply simp
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     apply blast
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    apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
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     prefer 2
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     apply (rule ext)
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     apply (simp (no_asm_simp))
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     apply (erule ssubst, rule premises, blast)
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    apply (erule allE, erule impE, erule_tac [2] mp, blast)
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    apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
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    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
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     prefer 2
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     apply blast
nipkow@11464
   322
    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
wenzelm@10249
   323
     prefer 2
wenzelm@10249
   324
     apply blast
nipkow@15316
   325
    apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
wenzelm@10249
   326
    done
wenzelm@10249
   327
qed
wenzelm@10249
   328
wenzelm@10313
   329
theorem rep_multiset_induct:
nipkow@11464
   330
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701
   331
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
wenzelm@17161
   332
  using rep_multiset_induct_aux by blast
wenzelm@10249
   333
wenzelm@10249
   334
theorem multiset_induct [induct type: multiset]:
wenzelm@17161
   335
  assumes prem1: "P {#}"
wenzelm@17161
   336
    and prem2: "!!M x. P M ==> P (M + {#x#})"
wenzelm@17161
   337
  shows "P M"
wenzelm@10249
   338
proof -
wenzelm@10249
   339
  note defns = union_def single_def Mempty_def
wenzelm@10249
   340
  show ?thesis
wenzelm@10249
   341
    apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313
   342
    apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@17161
   343
     apply (rule prem1 [unfolded defns])
paulson@15072
   344
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249
   345
     prefer 2
wenzelm@10249
   346
     apply (simp add: expand_fun_eq)
wenzelm@10249
   347
    apply (erule ssubst)
wenzelm@17200
   348
    apply (erule Abs_multiset_inverse [THEN subst])
wenzelm@17161
   349
    apply (erule prem2 [unfolded defns, simplified])
wenzelm@10249
   350
    done
wenzelm@10249
   351
qed
wenzelm@10249
   352
wenzelm@10249
   353
lemma MCollect_preserves_multiset:
nipkow@11464
   354
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
wenzelm@10249
   355
  apply (simp add: multiset_def)
paulson@15072
   356
  apply (rule finite_subset, auto)
wenzelm@10249
   357
  done
wenzelm@10249
   358
wenzelm@17161
   359
lemma count_MCollect [simp]:
wenzelm@10249
   360
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
paulson@15072
   361
  by (simp add: count_def MCollect_def MCollect_preserves_multiset)
wenzelm@10249
   362
wenzelm@17161
   363
lemma set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
wenzelm@17161
   364
  by (auto simp add: set_of_def)
wenzelm@10249
   365
wenzelm@17161
   366
lemma multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
wenzelm@17161
   367
  by (subst multiset_eq_conv_count_eq, auto)
wenzelm@10249
   368
wenzelm@17161
   369
lemma add_eq_conv_ex:
wenzelm@17161
   370
  "(M + {#a#} = N + {#b#}) =
wenzelm@17161
   371
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
paulson@15072
   372
  by (auto simp add: add_eq_conv_diff)
wenzelm@10249
   373
kleing@15869
   374
declare multiset_typedef [simp del]
wenzelm@10249
   375
wenzelm@17161
   376
wenzelm@10249
   377
subsection {* Multiset orderings *}
wenzelm@10249
   378
wenzelm@10249
   379
subsubsection {* Well-foundedness *}
wenzelm@10249
   380
wenzelm@10249
   381
constdefs
nipkow@11464
   382
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10249
   383
  "mult1 r ==
nipkow@11464
   384
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
nipkow@11464
   385
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249
   386
nipkow@11464
   387
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10392
   388
  "mult r == (mult1 r)\<^sup>+"
wenzelm@10249
   389
nipkow@11464
   390
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
wenzelm@10277
   391
  by (simp add: mult1_def)
wenzelm@10249
   392
nipkow@11464
   393
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
nipkow@11464
   394
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
nipkow@11464
   395
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
nipkow@11464
   396
  (concl is "?case1 (mult1 r) \<or> ?case2")
wenzelm@10249
   397
proof (unfold mult1_def)
nipkow@11464
   398
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   399
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
wenzelm@10249
   400
  let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249
   401
nipkow@11464
   402
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
nipkow@11464
   403
  hence "\<exists>a' M0' K.
nipkow@11464
   404
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
nipkow@11464
   405
  thus "?case1 \<or> ?case2"
wenzelm@10249
   406
  proof (elim exE conjE)
wenzelm@10249
   407
    fix a' M0' K
wenzelm@10249
   408
    assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249
   409
    assume "M0 + {#a#} = M0' + {#a'#}"
nipkow@11464
   410
    hence "M0 = M0' \<and> a = a' \<or>
nipkow@11464
   411
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249
   412
      by (simp only: add_eq_conv_ex)
wenzelm@10249
   413
    thus ?thesis
wenzelm@10249
   414
    proof (elim disjE conjE exE)
wenzelm@10249
   415
      assume "M0 = M0'" "a = a'"
nipkow@11464
   416
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@10249
   417
      hence ?case2 .. thus ?thesis ..
wenzelm@10249
   418
    next
wenzelm@10249
   419
      fix K'
wenzelm@10249
   420
      assume "M0' = K' + {#a#}"
wenzelm@10249
   421
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249
   422
wenzelm@10249
   423
      assume "M0 = K' + {#a'#}"
wenzelm@10249
   424
      with r have "?R (K' + K) M0" by blast
wenzelm@10249
   425
      with n have ?case1 by simp thus ?thesis ..
wenzelm@10249
   426
    qed
wenzelm@10249
   427
  qed
wenzelm@10249
   428
qed
wenzelm@10249
   429
nipkow@11464
   430
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249
   431
proof
wenzelm@10249
   432
  let ?R = "mult1 r"
wenzelm@10249
   433
  let ?W = "acc ?R"
wenzelm@10249
   434
  {
wenzelm@10249
   435
    fix M M0 a
nipkow@11464
   436
    assume M0: "M0 \<in> ?W"
wenzelm@12399
   437
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   438
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
nipkow@11464
   439
    have "M0 + {#a#} \<in> ?W"
wenzelm@10249
   440
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
   441
      fix N
nipkow@11464
   442
      assume "(N, M0 + {#a#}) \<in> ?R"
nipkow@11464
   443
      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
nipkow@11464
   444
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249
   445
        by (rule less_add)
nipkow@11464
   446
      thus "N \<in> ?W"
wenzelm@10249
   447
      proof (elim exE disjE conjE)
nipkow@11464
   448
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
nipkow@11464
   449
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
nipkow@11464
   450
        hence "M + {#a#} \<in> ?W" ..
nipkow@11464
   451
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   452
      next
wenzelm@10249
   453
        fix K
wenzelm@10249
   454
        assume N: "N = M0 + K"
nipkow@11464
   455
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   456
        have "?this --> M0 + K \<in> ?W" (is "?P K")
wenzelm@10249
   457
        proof (induct K)
nipkow@11464
   458
          from M0 have "M0 + {#} \<in> ?W" by simp
wenzelm@10249
   459
          thus "?P {#}" ..
wenzelm@10249
   460
wenzelm@10249
   461
          fix K x assume hyp: "?P K"
wenzelm@10249
   462
          show "?P (K + {#x#})"
wenzelm@10249
   463
          proof
nipkow@11464
   464
            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
nipkow@11464
   465
            hence "(x, a) \<in> r" by simp
nipkow@11464
   466
            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
wenzelm@10249
   467
nipkow@11464
   468
            from a hyp have "M0 + K \<in> ?W" by simp
nipkow@11464
   469
            with b have "(M0 + K) + {#x#} \<in> ?W" ..
nipkow@11464
   470
            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249
   471
          qed
wenzelm@10249
   472
        qed
nipkow@11464
   473
        hence "M0 + K \<in> ?W" ..
nipkow@11464
   474
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   475
      qed
wenzelm@10249
   476
    qed
wenzelm@10249
   477
  } note tedious_reasoning = this
wenzelm@10249
   478
wenzelm@10249
   479
  assume wf: "wf r"
wenzelm@10249
   480
  fix M
nipkow@11464
   481
  show "M \<in> ?W"
wenzelm@10249
   482
  proof (induct M)
nipkow@11464
   483
    show "{#} \<in> ?W"
wenzelm@10249
   484
    proof (rule accI)
nipkow@11464
   485
      fix b assume "(b, {#}) \<in> ?R"
nipkow@11464
   486
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
   487
    qed
wenzelm@10249
   488
nipkow@11464
   489
    fix M a assume "M \<in> ?W"
nipkow@11464
   490
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   491
    proof induct
wenzelm@10249
   492
      fix a
wenzelm@12399
   493
      assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   494
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   495
      proof
nipkow@11464
   496
        fix M assume "M \<in> ?W"
nipkow@11464
   497
        thus "M + {#a#} \<in> ?W"
wenzelm@10249
   498
          by (rule acc_induct) (rule tedious_reasoning)
wenzelm@10249
   499
      qed
wenzelm@10249
   500
    qed
nipkow@11464
   501
    thus "M + {#a#} \<in> ?W" ..
wenzelm@10249
   502
  qed
wenzelm@10249
   503
qed
wenzelm@10249
   504
wenzelm@10249
   505
theorem wf_mult1: "wf r ==> wf (mult1 r)"
wenzelm@10249
   506
  by (rule acc_wfI, rule all_accessible)
wenzelm@10249
   507
wenzelm@10249
   508
theorem wf_mult: "wf r ==> wf (mult r)"
wenzelm@10249
   509
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
wenzelm@10249
   510
wenzelm@10249
   511
wenzelm@10249
   512
subsubsection {* Closure-free presentation *}
wenzelm@10249
   513
wenzelm@10249
   514
(*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249
   515
wenzelm@10249
   516
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
paulson@15072
   517
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   518
wenzelm@10249
   519
text {* One direction. *}
wenzelm@10249
   520
wenzelm@10249
   521
lemma mult_implies_one_step:
nipkow@11464
   522
  "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464
   523
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
nipkow@11464
   524
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
wenzelm@10249
   525
  apply (unfold mult_def mult1_def set_of_def)
paulson@15072
   526
  apply (erule converse_trancl_induct, clarify)
paulson@15072
   527
   apply (rule_tac x = M0 in exI, simp, clarify)
wenzelm@10249
   528
  apply (case_tac "a :# K")
wenzelm@10249
   529
   apply (rule_tac x = I in exI)
wenzelm@10249
   530
   apply (simp (no_asm))
wenzelm@10249
   531
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
wenzelm@10249
   532
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@11464
   533
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
wenzelm@10249
   534
   apply (simp add: diff_union_single_conv)
wenzelm@10249
   535
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   536
   apply blast
wenzelm@10249
   537
  apply (subgoal_tac "a :# I")
wenzelm@10249
   538
   apply (rule_tac x = "I - {#a#}" in exI)
wenzelm@10249
   539
   apply (rule_tac x = "J + {#a#}" in exI)
wenzelm@10249
   540
   apply (rule_tac x = "K + Ka" in exI)
wenzelm@10249
   541
   apply (rule conjI)
wenzelm@10249
   542
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   543
   apply (rule conjI)
paulson@15072
   544
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
wenzelm@10249
   545
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   546
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   547
   apply blast
wenzelm@10277
   548
  apply (subgoal_tac "a :# (M0 + {#a#})")
wenzelm@10249
   549
   apply simp
wenzelm@10249
   550
  apply (simp (no_asm))
wenzelm@10249
   551
  done
wenzelm@10249
   552
wenzelm@10249
   553
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
paulson@15072
   554
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   555
nipkow@11464
   556
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
wenzelm@10249
   557
  apply (erule size_eq_Suc_imp_elem [THEN exE])
paulson@15072
   558
  apply (drule elem_imp_eq_diff_union, auto)
wenzelm@10249
   559
  done
wenzelm@10249
   560
wenzelm@10249
   561
lemma one_step_implies_mult_aux:
wenzelm@10249
   562
  "trans r ==>
nipkow@11464
   563
    \<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))
nipkow@11464
   564
      --> (I + K, I + J) \<in> mult r"
paulson@15072
   565
  apply (induct_tac n, auto)
paulson@15072
   566
  apply (frule size_eq_Suc_imp_eq_union, clarify)
paulson@15072
   567
  apply (rename_tac "J'", simp)
paulson@15072
   568
  apply (erule notE, auto)
wenzelm@10249
   569
  apply (case_tac "J' = {#}")
wenzelm@10249
   570
   apply (simp add: mult_def)
wenzelm@10249
   571
   apply (rule r_into_trancl)
paulson@15072
   572
   apply (simp add: mult1_def set_of_def, blast)
nipkow@11464
   573
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@11464
   574
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@11464
   575
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
wenzelm@10249
   576
  apply (erule ssubst)
paulson@15072
   577
  apply (simp add: Ball_def, auto)
wenzelm@10249
   578
  apply (subgoal_tac
nipkow@11464
   579
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
nipkow@11464
   580
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
wenzelm@10249
   581
   prefer 2
wenzelm@10249
   582
   apply force
wenzelm@10249
   583
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
wenzelm@10249
   584
  apply (erule trancl_trans)
wenzelm@10249
   585
  apply (rule r_into_trancl)
wenzelm@10249
   586
  apply (simp add: mult1_def set_of_def)
wenzelm@10249
   587
  apply (rule_tac x = a in exI)
wenzelm@10249
   588
  apply (rule_tac x = "I + J'" in exI)
wenzelm@10249
   589
  apply (simp add: union_ac)
wenzelm@10249
   590
  done
wenzelm@10249
   591
wenzelm@17161
   592
lemma one_step_implies_mult:
nipkow@11464
   593
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
nipkow@11464
   594
    ==> (I + K, I + J) \<in> mult r"
paulson@15072
   595
  apply (insert one_step_implies_mult_aux, blast)
wenzelm@10249
   596
  done
wenzelm@10249
   597
wenzelm@10249
   598
wenzelm@10249
   599
subsubsection {* Partial-order properties *}
wenzelm@10249
   600
wenzelm@12338
   601
instance multiset :: (type) ord ..
wenzelm@10249
   602
wenzelm@10249
   603
defs (overloaded)
nipkow@11464
   604
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
nipkow@11464
   605
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249
   606
wenzelm@10249
   607
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
wenzelm@10249
   608
  apply (unfold trans_def)
wenzelm@10249
   609
  apply (blast intro: order_less_trans)
wenzelm@10249
   610
  done
wenzelm@10249
   611
wenzelm@10249
   612
text {*
wenzelm@10249
   613
 \medskip Irreflexivity.
wenzelm@10249
   614
*}
wenzelm@10249
   615
wenzelm@10249
   616
lemma mult_irrefl_aux:
nipkow@11464
   617
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
wenzelm@10249
   618
  apply (erule finite_induct)
wenzelm@10249
   619
   apply (auto intro: order_less_trans)
wenzelm@10249
   620
  done
wenzelm@10249
   621
wenzelm@17161
   622
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
paulson@15072
   623
  apply (unfold less_multiset_def, auto)
paulson@15072
   624
  apply (drule trans_base_order [THEN mult_implies_one_step], auto)
wenzelm@10249
   625
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
wenzelm@10249
   626
  apply (simp add: set_of_eq_empty_iff)
wenzelm@10249
   627
  done
wenzelm@10249
   628
wenzelm@10249
   629
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
paulson@15072
   630
by (insert mult_less_not_refl, fast)
wenzelm@10249
   631
wenzelm@10249
   632
wenzelm@10249
   633
text {* Transitivity. *}
wenzelm@10249
   634
wenzelm@10249
   635
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
wenzelm@10249
   636
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   637
  apply (blast intro: trancl_trans)
wenzelm@10249
   638
  done
wenzelm@10249
   639
wenzelm@10249
   640
text {* Asymmetry. *}
wenzelm@10249
   641
nipkow@11464
   642
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
wenzelm@10249
   643
  apply auto
wenzelm@10249
   644
  apply (rule mult_less_not_refl [THEN notE])
paulson@15072
   645
  apply (erule mult_less_trans, assumption)
wenzelm@10249
   646
  done
wenzelm@10249
   647
wenzelm@10249
   648
theorem mult_less_asym:
nipkow@11464
   649
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
paulson@15072
   650
  by (insert mult_less_not_sym, blast)
wenzelm@10249
   651
wenzelm@10249
   652
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
paulson@15072
   653
by (unfold le_multiset_def, auto)
wenzelm@10249
   654
wenzelm@10249
   655
text {* Anti-symmetry. *}
wenzelm@10249
   656
wenzelm@10249
   657
theorem mult_le_antisym:
wenzelm@10249
   658
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
wenzelm@10249
   659
  apply (unfold le_multiset_def)
wenzelm@10249
   660
  apply (blast dest: mult_less_not_sym)
wenzelm@10249
   661
  done
wenzelm@10249
   662
wenzelm@10249
   663
text {* Transitivity. *}
wenzelm@10249
   664
wenzelm@10249
   665
theorem mult_le_trans:
wenzelm@10249
   666
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
wenzelm@10249
   667
  apply (unfold le_multiset_def)
wenzelm@10249
   668
  apply (blast intro: mult_less_trans)
wenzelm@10249
   669
  done
wenzelm@10249
   670
wenzelm@11655
   671
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
paulson@15072
   672
by (unfold le_multiset_def, auto)
wenzelm@10249
   673
wenzelm@10277
   674
text {* Partial order. *}
wenzelm@10277
   675
wenzelm@10277
   676
instance multiset :: (order) order
wenzelm@10277
   677
  apply intro_classes
wenzelm@10277
   678
     apply (rule mult_le_refl)
paulson@15072
   679
    apply (erule mult_le_trans, assumption)
paulson@15072
   680
   apply (erule mult_le_antisym, assumption)
wenzelm@10277
   681
  apply (rule mult_less_le)
wenzelm@10277
   682
  done
wenzelm@10277
   683
wenzelm@10249
   684
wenzelm@10249
   685
subsubsection {* Monotonicity of multiset union *}
wenzelm@10249
   686
wenzelm@17161
   687
lemma mult1_union:
nipkow@11464
   688
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
paulson@15072
   689
  apply (unfold mult1_def, auto)
wenzelm@10249
   690
  apply (rule_tac x = a in exI)
wenzelm@10249
   691
  apply (rule_tac x = "C + M0" in exI)
wenzelm@10249
   692
  apply (simp add: union_assoc)
wenzelm@10249
   693
  done
wenzelm@10249
   694
wenzelm@10249
   695
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
wenzelm@10249
   696
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   697
  apply (erule trancl_induct)
wenzelm@10249
   698
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
wenzelm@10249
   699
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
wenzelm@10249
   700
  done
wenzelm@10249
   701
wenzelm@10249
   702
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
wenzelm@10249
   703
  apply (subst union_commute [of B C])
wenzelm@10249
   704
  apply (subst union_commute [of D C])
wenzelm@10249
   705
  apply (erule union_less_mono2)
wenzelm@10249
   706
  done
wenzelm@10249
   707
wenzelm@17161
   708
lemma union_less_mono:
wenzelm@10249
   709
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
wenzelm@10249
   710
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249
   711
  done
wenzelm@10249
   712
wenzelm@17161
   713
lemma union_le_mono:
wenzelm@10249
   714
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
wenzelm@10249
   715
  apply (unfold le_multiset_def)
wenzelm@10249
   716
  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249
   717
  done
wenzelm@10249
   718
wenzelm@17161
   719
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
wenzelm@10249
   720
  apply (unfold le_multiset_def less_multiset_def)
wenzelm@10249
   721
  apply (case_tac "M = {#}")
wenzelm@10249
   722
   prefer 2
nipkow@11464
   723
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
wenzelm@10249
   724
    prefer 2
wenzelm@10249
   725
    apply (rule one_step_implies_mult)
paulson@15072
   726
      apply (simp only: trans_def, auto)
wenzelm@10249
   727
  done
wenzelm@10249
   728
wenzelm@17161
   729
lemma union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072
   730
proof -
wenzelm@17200
   731
  have "A + {#} <= A + B" by (blast intro: union_le_mono)
paulson@15072
   732
  thus ?thesis by simp
paulson@15072
   733
qed
paulson@15072
   734
wenzelm@17161
   735
lemma union_upper2: "B <= A + (B::'a::order multiset)"
paulson@15072
   736
by (subst union_commute, rule union_upper1)
paulson@15072
   737
paulson@15072
   738
wenzelm@17200
   739
subsection {* Link with lists *}
paulson@15072
   740
wenzelm@17200
   741
consts
paulson@15072
   742
  multiset_of :: "'a list \<Rightarrow> 'a multiset"
paulson@15072
   743
primrec
paulson@15072
   744
  "multiset_of [] = {#}"
paulson@15072
   745
  "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072
   746
paulson@15072
   747
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
wenzelm@17200
   748
  by (induct_tac x, auto)
paulson@15072
   749
paulson@15072
   750
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
paulson@15072
   751
  by (induct_tac x, auto)
paulson@15072
   752
paulson@15072
   753
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
wenzelm@17200
   754
  by (induct_tac x, auto)
kleing@15867
   755
kleing@15867
   756
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
kleing@15867
   757
  by (induct xs) auto
paulson@15072
   758
wenzelm@17200
   759
lemma multiset_of_append[simp]:
paulson@15072
   760
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
wenzelm@17200
   761
  by (rule_tac x=ys in spec, induct_tac xs, auto simp: union_ac)
paulson@15072
   762
paulson@15072
   763
lemma surj_multiset_of: "surj multiset_of"
wenzelm@17200
   764
  apply (unfold surj_def, rule allI)
wenzelm@17200
   765
  apply (rule_tac M=y in multiset_induct, auto)
wenzelm@17200
   766
  apply (rule_tac x = "x # xa" in exI, auto)
wenzelm@10249
   767
  done
wenzelm@10249
   768
paulson@15072
   769
lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
wenzelm@17200
   770
  by (induct_tac x, auto)
paulson@15072
   771
wenzelm@17200
   772
lemma distinct_count_atmost_1:
paulson@15072
   773
   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
paulson@15072
   774
   apply ( induct_tac x, simp, rule iffI, simp_all)
wenzelm@17200
   775
   apply (rule conjI)
wenzelm@17200
   776
   apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
paulson@15072
   777
   apply (erule_tac x=a in allE, simp, clarify)
wenzelm@17200
   778
   apply (erule_tac x=aa in allE, simp)
paulson@15072
   779
   done
paulson@15072
   780
wenzelm@17200
   781
lemma multiset_of_eq_setD:
kleing@15867
   782
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
kleing@15867
   783
  by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
kleing@15867
   784
wenzelm@17200
   785
lemma set_eq_iff_multiset_of_eq_distinct:
wenzelm@17200
   786
  "\<lbrakk>distinct x; distinct y\<rbrakk>
paulson@15072
   787
   \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
wenzelm@17200
   788
  by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
paulson@15072
   789
wenzelm@17200
   790
lemma set_eq_iff_multiset_of_remdups_eq:
paulson@15072
   791
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
wenzelm@17200
   792
  apply (rule iffI)
wenzelm@17200
   793
  apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
wenzelm@17200
   794
  apply (drule distinct_remdups[THEN distinct_remdups
wenzelm@17200
   795
                      [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]])
paulson@15072
   796
  apply simp
wenzelm@10249
   797
  done
wenzelm@10249
   798
kleing@15630
   799
lemma multiset_of_compl_union[simp]:
kleing@15630
   800
 "multiset_of [x\<in>xs. P x] + multiset_of [x\<in>xs. \<not>P x] = multiset_of xs"
kleing@15630
   801
  by (induct xs) (auto simp: union_ac)
paulson@15072
   802
wenzelm@17200
   803
lemma count_filter:
kleing@15867
   804
  "count (multiset_of xs) x = length [y \<in> xs. y = x]"
kleing@15867
   805
  by (induct xs, auto)
kleing@15867
   806
kleing@15867
   807
paulson@15072
   808
subsection {* Pointwise ordering induced by count *}
paulson@15072
   809
wenzelm@17200
   810
consts
paulson@15072
   811
  mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"
paulson@15072
   812
wenzelm@17200
   813
syntax
wenzelm@17200
   814
  "_mset_le" :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"   ("_ \<le># _"  [50,51] 50)
wenzelm@17200
   815
translations
paulson@15072
   816
  "x \<le># y" == "mset_le x y"
paulson@15072
   817
wenzelm@17200
   818
defs
wenzelm@17200
   819
  mset_le_def: "xs \<le># ys == (\<forall>a. count xs a \<le> count ys a)"
paulson@15072
   820
paulson@15072
   821
lemma mset_le_refl[simp]: "xs \<le># xs"
wenzelm@17200
   822
  by (unfold mset_le_def) auto
paulson@15072
   823
paulson@15072
   824
lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
wenzelm@17200
   825
  by (unfold mset_le_def) (fast intro: order_trans)
paulson@15072
   826
paulson@15072
   827
lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
wenzelm@17200
   828
  apply (unfold mset_le_def)
wenzelm@17200
   829
  apply (rule multiset_eq_conv_count_eq[THEN iffD2])
paulson@15072
   830
  apply (blast intro: order_antisym)
paulson@15072
   831
  done
paulson@15072
   832
wenzelm@17200
   833
lemma mset_le_exists_conv:
wenzelm@17200
   834
  "(xs \<le># ys) = (\<exists>zs. ys = xs + zs)"
wenzelm@17200
   835
  apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI)
paulson@15072
   836
  apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
paulson@15072
   837
  done
paulson@15072
   838
paulson@15072
   839
lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
wenzelm@17200
   840
  by (unfold mset_le_def) auto
paulson@15072
   841
paulson@15072
   842
lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
wenzelm@17200
   843
  by (unfold mset_le_def) auto
paulson@15072
   844
wenzelm@17200
   845
lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws"
wenzelm@17200
   846
  apply (unfold mset_le_def)
wenzelm@17200
   847
  apply auto
paulson@15072
   848
  apply (erule_tac x=a in allE)+
paulson@15072
   849
  apply auto
paulson@15072
   850
  done
paulson@15072
   851
paulson@15072
   852
lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
wenzelm@17200
   853
  by (unfold mset_le_def) auto
paulson@15072
   854
paulson@15072
   855
lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
wenzelm@17200
   856
  by (unfold mset_le_def) auto
paulson@15072
   857
paulson@15072
   858
lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
wenzelm@17200
   859
  apply (induct x)
wenzelm@17200
   860
   apply auto
wenzelm@17200
   861
  apply (rule mset_le_trans)
wenzelm@17200
   862
   apply auto
wenzelm@17200
   863
  done
paulson@15072
   864
wenzelm@10249
   865
end