src/HOL/Library/Multiset.thy
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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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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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e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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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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e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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parents:
diff changeset
   280
proof -
11549
e7265e70fd7c renamed "antecedent" case to "rule_context";
wenzelm
parents: 11464
diff changeset
   281
  case rule_context
e7265e70fd7c renamed "antecedent" case to "rule_context";
wenzelm
parents: 11464
diff changeset
   282
  note premises = this [unfolded multiset_def]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   283
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   284
    apply (unfold multiset_def)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   285
    apply (induct_tac n, simp, clarify)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   286
     apply (subgoal_tac "f = (\<lambda>a.0)")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   287
      apply simp
11549
e7265e70fd7c renamed "antecedent" case to "rule_context";
wenzelm
parents: 11464
diff changeset
   288
      apply (rule premises)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   289
     apply (rule ext, force, clarify)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   290
    apply (frule setsum_SucD, clarify)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   291
    apply (rename_tac a)
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   292
    apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   293
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   294
     apply (rule finite_subset)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   295
      prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   296
      apply assumption
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   297
     apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   298
     apply blast
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   299
    apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   300
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   301
     apply (rule ext)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   302
     apply (simp (no_asm_simp))
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   303
     apply (erule ssubst, rule premises, blast)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   304
    apply (erule allE, erule impE, erule_tac [2] mp, blast)
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   305
    apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   306
    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   307
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   308
     apply blast
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   309
    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   310
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   311
     apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   312
    apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   313
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   314
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   315
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   316
theorem rep_multiset_induct:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   317
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   318
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   319
  by (insert rep_multiset_induct_aux, blast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   320
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   321
theorem multiset_induct [induct type: multiset]:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   322
  "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   323
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   324
  note defns = union_def single_def Mempty_def
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   325
  assume prem1 [unfolded defns]: "P {#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   326
  assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   327
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   328
    apply (rule Rep_multiset_inverse [THEN subst])
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   329
    apply (rule Rep_multiset [THEN rep_multiset_induct])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   330
     apply (rule prem1)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   331
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   332
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   333
     apply (simp add: expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   334
    apply (erule ssubst)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   335
    apply (erule Abs_multiset_inverse [THEN subst])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   336
    apply (erule prem2 [simplified])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   337
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   338
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   339
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   340
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   341
lemma MCollect_preserves_multiset:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   342
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   343
  apply (simp add: multiset_def)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   344
  apply (rule finite_subset, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   345
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   346
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   347
theorem count_MCollect [simp]:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   348
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   349
  by (simp add: count_def MCollect_def MCollect_preserves_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   350
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   351
theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   352
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   353
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   354
theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   355
by (subst multiset_eq_conv_count_eq, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   356
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   357
declare Rep_multiset_inject [symmetric, simp del]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   358
declare multiset_typedef [simp del]
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   359
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   360
theorem add_eq_conv_ex:
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   361
      "(M + {#a#} = N + {#b#}) =
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   362
       (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   363
  by (auto simp add: add_eq_conv_diff)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   364
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   365
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   366
subsection {* Multiset orderings *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   367
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   368
subsubsection {* Well-foundedness *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   369
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   370
constdefs
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   371
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   372
  "mult1 r ==
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   373
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   374
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   375
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   376
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
10392
wenzelm
parents: 10313
diff changeset
   377
  "mult r == (mult1 r)\<^sup>+"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   378
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   379
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   380
  by (simp add: mult1_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   381
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   382
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   383
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   384
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   385
  (concl is "?case1 (mult1 r) \<or> ?case2")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   386
proof (unfold mult1_def)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   387
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   388
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   389
  let ?case1 = "?case1 {(N, M). ?R N M}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   390
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   391
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   392
  hence "\<exists>a' M0' K.
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   393
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   394
  thus "?case1 \<or> ?case2"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   395
  proof (elim exE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   396
    fix a' M0' K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   397
    assume N: "N = M0' + K" and r: "?r K a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   398
    assume "M0 + {#a#} = M0' + {#a'#}"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   399
    hence "M0 = M0' \<and> a = a' \<or>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   400
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   401
      by (simp only: add_eq_conv_ex)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   402
    thus ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   403
    proof (elim disjE conjE exE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   404
      assume "M0 = M0'" "a = a'"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   405
      with N r have "?r K a \<and> N = M0 + K" by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   406
      hence ?case2 .. thus ?thesis ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   407
    next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   408
      fix K'
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   409
      assume "M0' = K' + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   410
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   411
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   412
      assume "M0 = K' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   413
      with r have "?R (K' + K) M0" by blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   414
      with n have ?case1 by simp thus ?thesis ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   415
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   416
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   417
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   418
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   419
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   420
proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   421
  let ?R = "mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   422
  let ?W = "acc ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   423
  {
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   424
    fix M M0 a
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   425
    assume M0: "M0 \<in> ?W"
12399
wenzelm
parents: 12338
diff changeset
   426
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   427
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   428
    have "M0 + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   429
    proof (rule accI [of "M0 + {#a#}"])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   430
      fix N
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   431
      assume "(N, M0 + {#a#}) \<in> ?R"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   432
      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   433
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   434
        by (rule less_add)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   435
      thus "N \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   436
      proof (elim exE disjE conjE)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   437
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   438
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   439
        hence "M + {#a#} \<in> ?W" ..
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   440
        thus "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   441
      next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   442
        fix K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   443
        assume N: "N = M0 + K"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   444
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   445
        have "?this --> M0 + K \<in> ?W" (is "?P K")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   446
        proof (induct K)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   447
          from M0 have "M0 + {#} \<in> ?W" by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   448
          thus "?P {#}" ..
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   449
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   450
          fix K x assume hyp: "?P K"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   451
          show "?P (K + {#x#})"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   452
          proof
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   453
            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   454
            hence "(x, a) \<in> r" by simp
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   455
            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   456
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   457
            from a hyp have "M0 + K \<in> ?W" by simp
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   458
            with b have "(M0 + K) + {#x#} \<in> ?W" ..
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   459
            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   460
          qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   461
        qed
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   462
        hence "M0 + K \<in> ?W" ..
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   463
        thus "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   464
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   465
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   466
  } note tedious_reasoning = this
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   467
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   468
  assume wf: "wf r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   469
  fix M
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   470
  show "M \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   471
  proof (induct M)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   472
    show "{#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   473
    proof (rule accI)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   474
      fix b assume "(b, {#}) \<in> ?R"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   475
      with not_less_empty show "b \<in> ?W" by contradiction
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   476
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   477
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   478
    fix M a assume "M \<in> ?W"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   479
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   480
    proof induct
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   481
      fix a
12399
wenzelm
parents: 12338
diff changeset
   482
      assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   483
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   484
      proof
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   485
        fix M assume "M \<in> ?W"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   486
        thus "M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   487
          by (rule acc_induct) (rule tedious_reasoning)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   488
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   489
    qed
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   490
    thus "M + {#a#} \<in> ?W" ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   491
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   492
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   493
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   494
theorem wf_mult1: "wf r ==> wf (mult1 r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   495
  by (rule acc_wfI, rule all_accessible)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   496
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   497
theorem wf_mult: "wf r ==> wf (mult r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   498
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   499
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   500
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   501
subsubsection {* Closure-free presentation *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   502
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   503
(*Badly needed: a linear arithmetic procedure for multisets*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   504
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   505
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   506
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   507
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   508
text {* One direction. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   509
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   510
lemma mult_implies_one_step:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   511
  "trans r ==> (M, N) \<in> mult r ==>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   512
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   513
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   514
  apply (unfold mult_def mult1_def set_of_def)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   515
  apply (erule converse_trancl_induct, clarify)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   516
   apply (rule_tac x = M0 in exI, simp, clarify)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   517
  apply (case_tac "a :# K")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   518
   apply (rule_tac x = I in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   519
   apply (simp (no_asm))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   520
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   521
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   522
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   523
   apply (simp add: diff_union_single_conv)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   524
   apply (simp (no_asm_use) add: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   525
   apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   526
  apply (subgoal_tac "a :# I")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   527
   apply (rule_tac x = "I - {#a#}" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   528
   apply (rule_tac x = "J + {#a#}" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   529
   apply (rule_tac x = "K + Ka" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   530
   apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   531
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   532
   apply (rule conjI)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   533
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   534
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   535
   apply (simp (no_asm_use) add: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   536
   apply blast
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   537
  apply (subgoal_tac "a :# (M0 + {#a#})")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   538
   apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   539
  apply (simp (no_asm))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   540
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   541
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   542
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   543
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   544
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   545
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   546
  apply (erule size_eq_Suc_imp_elem [THEN exE])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   547
  apply (drule elem_imp_eq_diff_union, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   548
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   549
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   550
lemma one_step_implies_mult_aux:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   551
  "trans r ==>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   552
    \<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))
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   553
      --> (I + K, I + J) \<in> mult r"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   554
  apply (induct_tac n, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   555
  apply (frule size_eq_Suc_imp_eq_union, clarify)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   556
  apply (rename_tac "J'", simp)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   557
  apply (erule notE, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   558
  apply (case_tac "J' = {#}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   559
   apply (simp add: mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   560
   apply (rule r_into_trancl)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   561
   apply (simp add: mult1_def set_of_def, blast)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   562
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   563
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   564
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   565
  apply (erule ssubst)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   566
  apply (simp add: Ball_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   567
  apply (subgoal_tac
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   568
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   569
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   570
   prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   571
   apply force
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   572
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   573
  apply (erule trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   574
  apply (rule r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   575
  apply (simp add: mult1_def set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   576
  apply (rule_tac x = a in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   577
  apply (rule_tac x = "I + J'" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   578
  apply (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   579
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   580
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   581
theorem one_step_implies_mult:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   582
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   583
    ==> (I + K, I + J) \<in> mult r"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   584
  apply (insert one_step_implies_mult_aux, blast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   585
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   586
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   587
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   588
subsubsection {* Partial-order properties *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   589
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 11868
diff changeset
   590
instance multiset :: (type) ord ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   591
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   592
defs (overloaded)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   593
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   594
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   595
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   596
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   597
  apply (unfold trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   598
  apply (blast intro: order_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   599
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   600
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   601
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   602
 \medskip Irreflexivity.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   603
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   604
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   605
lemma mult_irrefl_aux:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   606
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   607
  apply (erule finite_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   608
   apply (auto intro: order_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   609
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   610
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   611
theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   612
  apply (unfold less_multiset_def, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   613
  apply (drule trans_base_order [THEN mult_implies_one_step], auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   614
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   615
  apply (simp add: set_of_eq_empty_iff)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   616
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   617
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   618
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   619
by (insert mult_less_not_refl, fast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   620
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   621
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   622
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   623
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   624
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   625
  apply (unfold less_multiset_def mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   626
  apply (blast intro: trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   627
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   628
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   629
text {* Asymmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   630
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   631
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   632
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   633
  apply (rule mult_less_not_refl [THEN notE])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   634
  apply (erule mult_less_trans, assumption)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   635
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   636
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   637
theorem mult_less_asym:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   638
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   639
  by (insert mult_less_not_sym, blast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   640
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   641
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   642
by (unfold le_multiset_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   643
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   644
text {* Anti-symmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   645
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   646
theorem mult_le_antisym:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   647
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   648
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   649
  apply (blast dest: mult_less_not_sym)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   650
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   651
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   652
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   653
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   654
theorem mult_le_trans:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   655
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   656
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   657
  apply (blast intro: mult_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   658
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   659
11655
923e4d0d36d5 tuned parentheses in relational expressions;
wenzelm
parents: 11549
diff changeset
   660
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   661
by (unfold le_multiset_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   662
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   663
text {* Partial order. *}
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   664
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   665
instance multiset :: (order) order
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   666
  apply intro_classes
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   667
     apply (rule mult_le_refl)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   668
    apply (erule mult_le_trans, assumption)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   669
   apply (erule mult_le_antisym, assumption)
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   670
  apply (rule mult_less_le)
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   671
  done
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   672
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   673
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   674
subsubsection {* Monotonicity of multiset union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   675
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   676
theorem mult1_union:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   677
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   678
  apply (unfold mult1_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   679
  apply (rule_tac x = a in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   680
  apply (rule_tac x = "C + M0" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   681
  apply (simp add: union_assoc)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   682
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   683
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   684
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   685
  apply (unfold less_multiset_def mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   686
  apply (erule trancl_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   687
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   688
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   689
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   690
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   691
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   692
  apply (subst union_commute [of B C])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   693
  apply (subst union_commute [of D C])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   694
  apply (erule union_less_mono2)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   695
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   696
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   697
theorem union_less_mono:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   698
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   699
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   700
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   701
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   702
theorem union_le_mono:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   703
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   704
  apply (unfold le_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   705
  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   706
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   707
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   708
theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   709
  apply (unfold le_multiset_def less_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   710
  apply (case_tac "M = {#}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   711
   prefer 2
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   712
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   713
    prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   714
    apply (rule one_step_implies_mult)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   715
      apply (simp only: trans_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   716
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   717
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   718
theorem union_upper1: "A <= A + (B::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   719
proof -
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   720
  have "A + {#} <= A + B" by (blast intro: union_le_mono) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   721
  thus ?thesis by simp
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   722
qed
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   723
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   724
theorem union_upper2: "B <= A + (B::'a::order multiset)"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   725
by (subst union_commute, rule union_upper1)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   726
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   727
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   728
subsection {* Link with lists *} 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   729
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   730
consts 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   731
  multiset_of :: "'a list \<Rightarrow> 'a multiset"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   732
primrec
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   733
  "multiset_of [] = {#}"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   734
  "multiset_of (a # x) = multiset_of x + {# a #}"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   735
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   736
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   737
  by (induct_tac x, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   738
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   739
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   740
  by (induct_tac x, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   741
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   742
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   743
 by (induct_tac x, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   744
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   745
lemma multset_of_append[simp]: 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   746
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   747
  by (rule_tac x=ys in spec, induct_tac xs, auto simp: union_ac) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   748
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   749
lemma surj_multiset_of: "surj multiset_of"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   750
  apply (unfold surj_def, rule allI) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   751
  apply (rule_tac M=y in multiset_induct, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   752
  apply (rule_tac x = "x # xa" in exI, auto) 
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   753
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   754
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   755
lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   756
  by (induct_tac x, auto)  
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   757
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   758
lemma distinct_count_atmost_1: 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   759
   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   760
   apply ( induct_tac x, simp, rule iffI, simp_all)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   761
   apply (rule conjI)  
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   762
   apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   763
   apply (erule_tac x=a in allE, simp, clarify)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   764
   apply (erule_tac x=aa in allE, simp) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   765
   done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   766
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   767
lemma set_eq_iff_multiset_of_eq_distinct: 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   768
  "\<lbrakk>distinct x; distinct y\<rbrakk> 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   769
   \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   770
  by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   771
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   772
lemma set_eq_iff_multiset_of_remdups_eq: 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   773
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   774
  apply (rule iffI) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   775
  apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   776
  apply (drule distinct_remdups[THEN distinct_remdups 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   777
                      [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]]) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   778
  apply simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   779
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   780
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   781
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   782
subsection {* Pointwise ordering induced by count *}
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   783
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   784
consts 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   785
  mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   786
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   787
syntax 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   788
  "_mset_le" :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"   ("_ \<le># _"  [50,51] 50) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   789
translations 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   790
  "x \<le># y" == "mset_le x y"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   791
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   792
defs 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   793
  mset_le_def:   "xs \<le># ys  == (! a. count xs a \<le> count ys a)"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   794
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   795
lemma mset_le_refl[simp]: "xs \<le># xs"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   796
  by (unfold mset_le_def, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   797
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   798
lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   799
  by (unfold mset_le_def, fast intro: order_trans) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   800
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   801
lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   802
  apply (unfold mset_le_def) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   803
  apply (rule multiset_eq_conv_count_eq[THEN iffD2]) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   804
  apply (blast intro: order_antisym)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   805
  done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   806
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   807
lemma mset_le_exists_conv: 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   808
  "(xs \<le># ys) = (? zs. ys = xs + zs)"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   809
  apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   810
  apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   811
  done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   812
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   813
lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   814
  by (unfold mset_le_def, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   815
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   816
lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   817
  by (unfold mset_le_def, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   818
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   819
lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws" 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   820
  apply (unfold mset_le_def, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   821
  apply (erule_tac x=a in allE)+
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   822
  apply auto
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   823
  done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   824
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   825
lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   826
  by (unfold mset_le_def, auto) 
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   827
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   828
lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   829
  by (unfold mset_le_def, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   830
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   831
lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   832
  by (induct_tac x, auto, rule mset_le_trans, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   833
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   834
end