src/HOL/Library/Multiset.thy
author paulson
Thu Jul 22 10:33:26 2004 +0200 (2004-07-22)
changeset 15072 4861bf6af0b4
parent 14981 e73f8140af78
child 15131 c69542757a4d
permissions -rw-r--r--
new material courtesy of Norbert Voelker
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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 = Accessible_Part:
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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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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 (unfold multiset_def, simp)
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  apply (drule finite_UnI, assumption)
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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 (unfold multiset_def, simp)
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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 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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theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
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by (simp add: union_def Mempty_def)
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theorem union_commute: "M + N = N + (M::'a multiset)"
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by (simp add: union_def add_ac)
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theorem 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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theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
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  apply (rule union_commute [THEN trans])
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  apply (rule union_assoc [THEN trans])
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  apply (rule union_commute [THEN arg_cong])
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  done
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theorems 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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theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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by (simp add: Mempty_def diff_def)
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theorem 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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theorem count_empty [simp]: "count {#} a = 0"
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by (simp add: count_def Mempty_def)
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theorem 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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theorem 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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theorem 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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theorem set_of_empty [simp]: "set_of {#} = {}"
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by (simp add: set_of_def)
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theorem set_of_single [simp]: "set_of {#b#} = {b}"
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by (simp add: set_of_def)
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theorem 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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theorem 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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theorem 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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theorem size_empty [simp]: "size {#} = 0"
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by (simp add: size_def)
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theorem size_single [simp]: "size {#b#} = 1"
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by (simp add: size_def)
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theorem finite_set_of [iff]: "finite (set_of M)"
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  apply (cut_tac x = M in Rep_multiset)
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  apply (simp add: multiset_def set_of_def count_def)
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  done
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theorem 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, simp)
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  apply (simp add: Int_insert_left set_of_def)
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  done
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theorem 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 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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(*
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val prems = Goal
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 "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
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by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
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     measure_induct 1);
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by (Clarify_tac 1)
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by (resolve_tac prems 1)
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 by (assume_tac 1)
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by (Clarify_tac 1)
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by (subgoal_tac "finite G" 1)
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 by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
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by (etac allE 1)
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by (etac impE 1)
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 by (Blast_tac 2)
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by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
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no_qed();
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val lemma = result();
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val prems = Goal
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 "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
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by (rtac (lemma RS mp) 1);
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by (REPEAT(ares_tac prems 1));
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qed "finite_psubset_induct";
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Better: use wf_finite_psubset in WF_Rel
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*)
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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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  "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
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    ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
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proof -
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  case rule_context
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  note premises = this [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
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    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
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     prefer 2
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     apply blast
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    apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
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    done
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qed
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theorem rep_multiset_induct:
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  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
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    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
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  by (insert rep_multiset_induct_aux, blast)
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theorem multiset_induct [induct type: multiset]:
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  "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
wenzelm@10249
   321
proof -
wenzelm@10249
   322
  note defns = union_def single_def Mempty_def
wenzelm@10249
   323
  assume prem1 [unfolded defns]: "P {#}"
wenzelm@10249
   324
  assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
wenzelm@10249
   325
  show ?thesis
wenzelm@10249
   326
    apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313
   327
    apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@10249
   328
     apply (rule prem1)
paulson@15072
   329
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249
   330
     prefer 2
wenzelm@10249
   331
     apply (simp add: expand_fun_eq)
wenzelm@10249
   332
    apply (erule ssubst)
wenzelm@10249
   333
    apply (erule Abs_multiset_inverse [THEN subst])
wenzelm@10249
   334
    apply (erule prem2 [simplified])
wenzelm@10249
   335
    done
wenzelm@10249
   336
qed
wenzelm@10249
   337
wenzelm@10249
   338
wenzelm@10249
   339
lemma MCollect_preserves_multiset:
nipkow@11464
   340
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
wenzelm@10249
   341
  apply (simp add: multiset_def)
paulson@15072
   342
  apply (rule finite_subset, auto)
wenzelm@10249
   343
  done
wenzelm@10249
   344
wenzelm@10249
   345
theorem count_MCollect [simp]:
wenzelm@10249
   346
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
paulson@15072
   347
  by (simp add: count_def MCollect_def MCollect_preserves_multiset)
wenzelm@10249
   348
nipkow@11464
   349
theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
paulson@15072
   350
by (auto simp add: set_of_def)
wenzelm@10249
   351
nipkow@11464
   352
theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
paulson@15072
   353
by (subst multiset_eq_conv_count_eq, auto)
wenzelm@10249
   354
wenzelm@10277
   355
declare Rep_multiset_inject [symmetric, simp del]
wenzelm@10249
   356
declare multiset_typedef [simp del]
wenzelm@10249
   357
wenzelm@10249
   358
theorem add_eq_conv_ex:
paulson@15072
   359
      "(M + {#a#} = N + {#b#}) =
paulson@15072
   360
       (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
paulson@15072
   361
  by (auto simp add: add_eq_conv_diff)
wenzelm@10249
   362
wenzelm@10249
   363
wenzelm@10249
   364
subsection {* Multiset orderings *}
wenzelm@10249
   365
wenzelm@10249
   366
subsubsection {* Well-foundedness *}
wenzelm@10249
   367
wenzelm@10249
   368
constdefs
nipkow@11464
   369
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10249
   370
  "mult1 r ==
nipkow@11464
   371
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
nipkow@11464
   372
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249
   373
nipkow@11464
   374
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10392
   375
  "mult r == (mult1 r)\<^sup>+"
wenzelm@10249
   376
nipkow@11464
   377
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
wenzelm@10277
   378
  by (simp add: mult1_def)
wenzelm@10249
   379
nipkow@11464
   380
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
nipkow@11464
   381
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
nipkow@11464
   382
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
nipkow@11464
   383
  (concl is "?case1 (mult1 r) \<or> ?case2")
wenzelm@10249
   384
proof (unfold mult1_def)
nipkow@11464
   385
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   386
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
wenzelm@10249
   387
  let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249
   388
nipkow@11464
   389
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
nipkow@11464
   390
  hence "\<exists>a' M0' K.
nipkow@11464
   391
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
nipkow@11464
   392
  thus "?case1 \<or> ?case2"
wenzelm@10249
   393
  proof (elim exE conjE)
wenzelm@10249
   394
    fix a' M0' K
wenzelm@10249
   395
    assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249
   396
    assume "M0 + {#a#} = M0' + {#a'#}"
nipkow@11464
   397
    hence "M0 = M0' \<and> a = a' \<or>
nipkow@11464
   398
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249
   399
      by (simp only: add_eq_conv_ex)
wenzelm@10249
   400
    thus ?thesis
wenzelm@10249
   401
    proof (elim disjE conjE exE)
wenzelm@10249
   402
      assume "M0 = M0'" "a = a'"
nipkow@11464
   403
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@10249
   404
      hence ?case2 .. thus ?thesis ..
wenzelm@10249
   405
    next
wenzelm@10249
   406
      fix K'
wenzelm@10249
   407
      assume "M0' = K' + {#a#}"
wenzelm@10249
   408
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249
   409
wenzelm@10249
   410
      assume "M0 = K' + {#a'#}"
wenzelm@10249
   411
      with r have "?R (K' + K) M0" by blast
wenzelm@10249
   412
      with n have ?case1 by simp thus ?thesis ..
wenzelm@10249
   413
    qed
wenzelm@10249
   414
  qed
wenzelm@10249
   415
qed
wenzelm@10249
   416
nipkow@11464
   417
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249
   418
proof
wenzelm@10249
   419
  let ?R = "mult1 r"
wenzelm@10249
   420
  let ?W = "acc ?R"
wenzelm@10249
   421
  {
wenzelm@10249
   422
    fix M M0 a
nipkow@11464
   423
    assume M0: "M0 \<in> ?W"
wenzelm@12399
   424
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   425
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
nipkow@11464
   426
    have "M0 + {#a#} \<in> ?W"
wenzelm@10249
   427
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
   428
      fix N
nipkow@11464
   429
      assume "(N, M0 + {#a#}) \<in> ?R"
nipkow@11464
   430
      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
nipkow@11464
   431
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249
   432
        by (rule less_add)
nipkow@11464
   433
      thus "N \<in> ?W"
wenzelm@10249
   434
      proof (elim exE disjE conjE)
nipkow@11464
   435
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
nipkow@11464
   436
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
nipkow@11464
   437
        hence "M + {#a#} \<in> ?W" ..
nipkow@11464
   438
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   439
      next
wenzelm@10249
   440
        fix K
wenzelm@10249
   441
        assume N: "N = M0 + K"
nipkow@11464
   442
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   443
        have "?this --> M0 + K \<in> ?W" (is "?P K")
wenzelm@10249
   444
        proof (induct K)
nipkow@11464
   445
          from M0 have "M0 + {#} \<in> ?W" by simp
wenzelm@10249
   446
          thus "?P {#}" ..
wenzelm@10249
   447
wenzelm@10249
   448
          fix K x assume hyp: "?P K"
wenzelm@10249
   449
          show "?P (K + {#x#})"
wenzelm@10249
   450
          proof
nipkow@11464
   451
            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
nipkow@11464
   452
            hence "(x, a) \<in> r" by simp
nipkow@11464
   453
            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
wenzelm@10249
   454
nipkow@11464
   455
            from a hyp have "M0 + K \<in> ?W" by simp
nipkow@11464
   456
            with b have "(M0 + K) + {#x#} \<in> ?W" ..
nipkow@11464
   457
            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249
   458
          qed
wenzelm@10249
   459
        qed
nipkow@11464
   460
        hence "M0 + K \<in> ?W" ..
nipkow@11464
   461
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   462
      qed
wenzelm@10249
   463
    qed
wenzelm@10249
   464
  } note tedious_reasoning = this
wenzelm@10249
   465
wenzelm@10249
   466
  assume wf: "wf r"
wenzelm@10249
   467
  fix M
nipkow@11464
   468
  show "M \<in> ?W"
wenzelm@10249
   469
  proof (induct M)
nipkow@11464
   470
    show "{#} \<in> ?W"
wenzelm@10249
   471
    proof (rule accI)
nipkow@11464
   472
      fix b assume "(b, {#}) \<in> ?R"
nipkow@11464
   473
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
   474
    qed
wenzelm@10249
   475
nipkow@11464
   476
    fix M a assume "M \<in> ?W"
nipkow@11464
   477
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   478
    proof induct
wenzelm@10249
   479
      fix a
wenzelm@12399
   480
      assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   481
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   482
      proof
nipkow@11464
   483
        fix M assume "M \<in> ?W"
nipkow@11464
   484
        thus "M + {#a#} \<in> ?W"
wenzelm@10249
   485
          by (rule acc_induct) (rule tedious_reasoning)
wenzelm@10249
   486
      qed
wenzelm@10249
   487
    qed
nipkow@11464
   488
    thus "M + {#a#} \<in> ?W" ..
wenzelm@10249
   489
  qed
wenzelm@10249
   490
qed
wenzelm@10249
   491
wenzelm@10249
   492
theorem wf_mult1: "wf r ==> wf (mult1 r)"
wenzelm@10249
   493
  by (rule acc_wfI, rule all_accessible)
wenzelm@10249
   494
wenzelm@10249
   495
theorem wf_mult: "wf r ==> wf (mult r)"
wenzelm@10249
   496
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
wenzelm@10249
   497
wenzelm@10249
   498
wenzelm@10249
   499
subsubsection {* Closure-free presentation *}
wenzelm@10249
   500
wenzelm@10249
   501
(*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249
   502
wenzelm@10249
   503
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
paulson@15072
   504
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   505
wenzelm@10249
   506
text {* One direction. *}
wenzelm@10249
   507
wenzelm@10249
   508
lemma mult_implies_one_step:
nipkow@11464
   509
  "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464
   510
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
nipkow@11464
   511
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
wenzelm@10249
   512
  apply (unfold mult_def mult1_def set_of_def)
paulson@15072
   513
  apply (erule converse_trancl_induct, clarify)
paulson@15072
   514
   apply (rule_tac x = M0 in exI, simp, clarify)
wenzelm@10249
   515
  apply (case_tac "a :# K")
wenzelm@10249
   516
   apply (rule_tac x = I in exI)
wenzelm@10249
   517
   apply (simp (no_asm))
wenzelm@10249
   518
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
wenzelm@10249
   519
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@11464
   520
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
wenzelm@10249
   521
   apply (simp add: diff_union_single_conv)
wenzelm@10249
   522
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   523
   apply blast
wenzelm@10249
   524
  apply (subgoal_tac "a :# I")
wenzelm@10249
   525
   apply (rule_tac x = "I - {#a#}" in exI)
wenzelm@10249
   526
   apply (rule_tac x = "J + {#a#}" in exI)
wenzelm@10249
   527
   apply (rule_tac x = "K + Ka" in exI)
wenzelm@10249
   528
   apply (rule conjI)
wenzelm@10249
   529
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   530
   apply (rule conjI)
paulson@15072
   531
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
wenzelm@10249
   532
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   533
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   534
   apply blast
wenzelm@10277
   535
  apply (subgoal_tac "a :# (M0 + {#a#})")
wenzelm@10249
   536
   apply simp
wenzelm@10249
   537
  apply (simp (no_asm))
wenzelm@10249
   538
  done
wenzelm@10249
   539
wenzelm@10249
   540
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
paulson@15072
   541
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   542
nipkow@11464
   543
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
wenzelm@10249
   544
  apply (erule size_eq_Suc_imp_elem [THEN exE])
paulson@15072
   545
  apply (drule elem_imp_eq_diff_union, auto)
wenzelm@10249
   546
  done
wenzelm@10249
   547
wenzelm@10249
   548
lemma one_step_implies_mult_aux:
wenzelm@10249
   549
  "trans r ==>
nipkow@11464
   550
    \<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
   551
      --> (I + K, I + J) \<in> mult r"
paulson@15072
   552
  apply (induct_tac n, auto)
paulson@15072
   553
  apply (frule size_eq_Suc_imp_eq_union, clarify)
paulson@15072
   554
  apply (rename_tac "J'", simp)
paulson@15072
   555
  apply (erule notE, auto)
wenzelm@10249
   556
  apply (case_tac "J' = {#}")
wenzelm@10249
   557
   apply (simp add: mult_def)
wenzelm@10249
   558
   apply (rule r_into_trancl)
paulson@15072
   559
   apply (simp add: mult1_def set_of_def, blast)
nipkow@11464
   560
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@11464
   561
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@11464
   562
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
wenzelm@10249
   563
  apply (erule ssubst)
paulson@15072
   564
  apply (simp add: Ball_def, auto)
wenzelm@10249
   565
  apply (subgoal_tac
nipkow@11464
   566
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
nipkow@11464
   567
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
wenzelm@10249
   568
   prefer 2
wenzelm@10249
   569
   apply force
wenzelm@10249
   570
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
wenzelm@10249
   571
  apply (erule trancl_trans)
wenzelm@10249
   572
  apply (rule r_into_trancl)
wenzelm@10249
   573
  apply (simp add: mult1_def set_of_def)
wenzelm@10249
   574
  apply (rule_tac x = a in exI)
wenzelm@10249
   575
  apply (rule_tac x = "I + J'" in exI)
wenzelm@10249
   576
  apply (simp add: union_ac)
wenzelm@10249
   577
  done
wenzelm@10249
   578
wenzelm@10249
   579
theorem one_step_implies_mult:
nipkow@11464
   580
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
nipkow@11464
   581
    ==> (I + K, I + J) \<in> mult r"
paulson@15072
   582
  apply (insert one_step_implies_mult_aux, blast)
wenzelm@10249
   583
  done
wenzelm@10249
   584
wenzelm@10249
   585
wenzelm@10249
   586
subsubsection {* Partial-order properties *}
wenzelm@10249
   587
wenzelm@12338
   588
instance multiset :: (type) ord ..
wenzelm@10249
   589
wenzelm@10249
   590
defs (overloaded)
nipkow@11464
   591
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
nipkow@11464
   592
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249
   593
wenzelm@10249
   594
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
wenzelm@10249
   595
  apply (unfold trans_def)
wenzelm@10249
   596
  apply (blast intro: order_less_trans)
wenzelm@10249
   597
  done
wenzelm@10249
   598
wenzelm@10249
   599
text {*
wenzelm@10249
   600
 \medskip Irreflexivity.
wenzelm@10249
   601
*}
wenzelm@10249
   602
wenzelm@10249
   603
lemma mult_irrefl_aux:
nipkow@11464
   604
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
wenzelm@10249
   605
  apply (erule finite_induct)
wenzelm@10249
   606
   apply (auto intro: order_less_trans)
wenzelm@10249
   607
  done
wenzelm@10249
   608
nipkow@11464
   609
theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
paulson@15072
   610
  apply (unfold less_multiset_def, auto)
paulson@15072
   611
  apply (drule trans_base_order [THEN mult_implies_one_step], auto)
wenzelm@10249
   612
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
wenzelm@10249
   613
  apply (simp add: set_of_eq_empty_iff)
wenzelm@10249
   614
  done
wenzelm@10249
   615
wenzelm@10249
   616
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
paulson@15072
   617
by (insert mult_less_not_refl, fast)
wenzelm@10249
   618
wenzelm@10249
   619
wenzelm@10249
   620
text {* Transitivity. *}
wenzelm@10249
   621
wenzelm@10249
   622
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
wenzelm@10249
   623
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   624
  apply (blast intro: trancl_trans)
wenzelm@10249
   625
  done
wenzelm@10249
   626
wenzelm@10249
   627
text {* Asymmetry. *}
wenzelm@10249
   628
nipkow@11464
   629
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
wenzelm@10249
   630
  apply auto
wenzelm@10249
   631
  apply (rule mult_less_not_refl [THEN notE])
paulson@15072
   632
  apply (erule mult_less_trans, assumption)
wenzelm@10249
   633
  done
wenzelm@10249
   634
wenzelm@10249
   635
theorem mult_less_asym:
nipkow@11464
   636
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
paulson@15072
   637
  by (insert mult_less_not_sym, blast)
wenzelm@10249
   638
wenzelm@10249
   639
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
paulson@15072
   640
by (unfold le_multiset_def, auto)
wenzelm@10249
   641
wenzelm@10249
   642
text {* Anti-symmetry. *}
wenzelm@10249
   643
wenzelm@10249
   644
theorem mult_le_antisym:
wenzelm@10249
   645
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
wenzelm@10249
   646
  apply (unfold le_multiset_def)
wenzelm@10249
   647
  apply (blast dest: mult_less_not_sym)
wenzelm@10249
   648
  done
wenzelm@10249
   649
wenzelm@10249
   650
text {* Transitivity. *}
wenzelm@10249
   651
wenzelm@10249
   652
theorem mult_le_trans:
wenzelm@10249
   653
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
wenzelm@10249
   654
  apply (unfold le_multiset_def)
wenzelm@10249
   655
  apply (blast intro: mult_less_trans)
wenzelm@10249
   656
  done
wenzelm@10249
   657
wenzelm@11655
   658
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
paulson@15072
   659
by (unfold le_multiset_def, auto)
wenzelm@10249
   660
wenzelm@10277
   661
text {* Partial order. *}
wenzelm@10277
   662
wenzelm@10277
   663
instance multiset :: (order) order
wenzelm@10277
   664
  apply intro_classes
wenzelm@10277
   665
     apply (rule mult_le_refl)
paulson@15072
   666
    apply (erule mult_le_trans, assumption)
paulson@15072
   667
   apply (erule mult_le_antisym, assumption)
wenzelm@10277
   668
  apply (rule mult_less_le)
wenzelm@10277
   669
  done
wenzelm@10277
   670
wenzelm@10249
   671
wenzelm@10249
   672
subsubsection {* Monotonicity of multiset union *}
wenzelm@10249
   673
wenzelm@10249
   674
theorem mult1_union:
nipkow@11464
   675
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
paulson@15072
   676
  apply (unfold mult1_def, auto)
wenzelm@10249
   677
  apply (rule_tac x = a in exI)
wenzelm@10249
   678
  apply (rule_tac x = "C + M0" in exI)
wenzelm@10249
   679
  apply (simp add: union_assoc)
wenzelm@10249
   680
  done
wenzelm@10249
   681
wenzelm@10249
   682
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
wenzelm@10249
   683
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   684
  apply (erule trancl_induct)
wenzelm@10249
   685
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
wenzelm@10249
   686
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
wenzelm@10249
   687
  done
wenzelm@10249
   688
wenzelm@10249
   689
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
wenzelm@10249
   690
  apply (subst union_commute [of B C])
wenzelm@10249
   691
  apply (subst union_commute [of D C])
wenzelm@10249
   692
  apply (erule union_less_mono2)
wenzelm@10249
   693
  done
wenzelm@10249
   694
wenzelm@10249
   695
theorem union_less_mono:
wenzelm@10249
   696
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
wenzelm@10249
   697
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249
   698
  done
wenzelm@10249
   699
wenzelm@10249
   700
theorem union_le_mono:
wenzelm@10249
   701
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
wenzelm@10249
   702
  apply (unfold le_multiset_def)
wenzelm@10249
   703
  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249
   704
  done
wenzelm@10249
   705
wenzelm@10249
   706
theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
wenzelm@10249
   707
  apply (unfold le_multiset_def less_multiset_def)
wenzelm@10249
   708
  apply (case_tac "M = {#}")
wenzelm@10249
   709
   prefer 2
nipkow@11464
   710
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
wenzelm@10249
   711
    prefer 2
wenzelm@10249
   712
    apply (rule one_step_implies_mult)
paulson@15072
   713
      apply (simp only: trans_def, auto)
wenzelm@10249
   714
  done
wenzelm@10249
   715
wenzelm@10249
   716
theorem union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072
   717
proof -
paulson@15072
   718
  have "A + {#} <= A + B" by (blast intro: union_le_mono) 
paulson@15072
   719
  thus ?thesis by simp
paulson@15072
   720
qed
paulson@15072
   721
paulson@15072
   722
theorem union_upper2: "B <= A + (B::'a::order multiset)"
paulson@15072
   723
by (subst union_commute, rule union_upper1)
paulson@15072
   724
paulson@15072
   725
paulson@15072
   726
subsection {* Link with lists *} 
paulson@15072
   727
paulson@15072
   728
consts 
paulson@15072
   729
  multiset_of :: "'a list \<Rightarrow> 'a multiset"
paulson@15072
   730
primrec
paulson@15072
   731
  "multiset_of [] = {#}"
paulson@15072
   732
  "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072
   733
paulson@15072
   734
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
paulson@15072
   735
  by (induct_tac x, auto) 
paulson@15072
   736
paulson@15072
   737
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
paulson@15072
   738
  by (induct_tac x, auto)
paulson@15072
   739
paulson@15072
   740
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
paulson@15072
   741
 by (induct_tac x, auto) 
paulson@15072
   742
paulson@15072
   743
lemma multset_of_append[simp]: 
paulson@15072
   744
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
paulson@15072
   745
  by (rule_tac x=ys in spec, induct_tac xs, auto simp: union_ac) 
paulson@15072
   746
paulson@15072
   747
lemma surj_multiset_of: "surj multiset_of"
paulson@15072
   748
  apply (unfold surj_def, rule allI) 
paulson@15072
   749
  apply (rule_tac M=y in multiset_induct, auto) 
paulson@15072
   750
  apply (rule_tac x = "x # xa" in exI, auto) 
wenzelm@10249
   751
  done
wenzelm@10249
   752
paulson@15072
   753
lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
paulson@15072
   754
  by (induct_tac x, auto)  
paulson@15072
   755
paulson@15072
   756
lemma distinct_count_atmost_1: 
paulson@15072
   757
   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
paulson@15072
   758
   apply ( induct_tac x, simp, rule iffI, simp_all)
paulson@15072
   759
   apply (rule conjI)  
paulson@15072
   760
   apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) 
paulson@15072
   761
   apply (erule_tac x=a in allE, simp, clarify)
paulson@15072
   762
   apply (erule_tac x=aa in allE, simp) 
paulson@15072
   763
   done
paulson@15072
   764
paulson@15072
   765
lemma set_eq_iff_multiset_of_eq_distinct: 
paulson@15072
   766
  "\<lbrakk>distinct x; distinct y\<rbrakk> 
paulson@15072
   767
   \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
paulson@15072
   768
  by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) 
paulson@15072
   769
paulson@15072
   770
lemma set_eq_iff_multiset_of_remdups_eq: 
paulson@15072
   771
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
paulson@15072
   772
  apply (rule iffI) 
paulson@15072
   773
  apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) 
paulson@15072
   774
  apply (drule distinct_remdups[THEN distinct_remdups 
paulson@15072
   775
                      [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]]) 
paulson@15072
   776
  apply simp
wenzelm@10249
   777
  done
wenzelm@10249
   778
paulson@15072
   779
paulson@15072
   780
subsection {* Pointwise ordering induced by count *}
paulson@15072
   781
paulson@15072
   782
consts 
paulson@15072
   783
  mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"
paulson@15072
   784
paulson@15072
   785
syntax 
paulson@15072
   786
  "_mset_le" :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"   ("_ \<le># _"  [50,51] 50) 
paulson@15072
   787
translations 
paulson@15072
   788
  "x \<le># y" == "mset_le x y"
paulson@15072
   789
paulson@15072
   790
defs 
paulson@15072
   791
  mset_le_def:   "xs \<le># ys  == (! a. count xs a \<le> count ys a)"
paulson@15072
   792
paulson@15072
   793
lemma mset_le_refl[simp]: "xs \<le># xs"
paulson@15072
   794
  by (unfold mset_le_def, auto) 
paulson@15072
   795
paulson@15072
   796
lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
paulson@15072
   797
  by (unfold mset_le_def, fast intro: order_trans) 
paulson@15072
   798
paulson@15072
   799
lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
paulson@15072
   800
  apply (unfold mset_le_def) 
paulson@15072
   801
  apply (rule multiset_eq_conv_count_eq[THEN iffD2]) 
paulson@15072
   802
  apply (blast intro: order_antisym)
paulson@15072
   803
  done
paulson@15072
   804
paulson@15072
   805
lemma mset_le_exists_conv: 
paulson@15072
   806
  "(xs \<le># ys) = (? zs. ys = xs + zs)"
paulson@15072
   807
  apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI) 
paulson@15072
   808
  apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
paulson@15072
   809
  done
paulson@15072
   810
paulson@15072
   811
lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
paulson@15072
   812
  by (unfold mset_le_def, auto) 
paulson@15072
   813
paulson@15072
   814
lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
paulson@15072
   815
  by (unfold mset_le_def, auto) 
paulson@15072
   816
paulson@15072
   817
lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws" 
paulson@15072
   818
  apply (unfold mset_le_def, auto) 
paulson@15072
   819
  apply (erule_tac x=a in allE)+
paulson@15072
   820
  apply auto
paulson@15072
   821
  done
paulson@15072
   822
paulson@15072
   823
lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
paulson@15072
   824
  by (unfold mset_le_def, auto) 
paulson@15072
   825
paulson@15072
   826
lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
paulson@15072
   827
  by (unfold mset_le_def, auto)
paulson@15072
   828
paulson@15072
   829
lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
paulson@15072
   830
  by (induct_tac x, auto, rule mset_le_trans, auto)
paulson@15072
   831
wenzelm@10249
   832
end