src/HOL/Library/Multiset.thy
author chaieb
Mon Feb 09 17:21:46 2009 +0000 (2009-02-09)
changeset 29847 af32126ee729
parent 29509 1ff0f3f08a7b
child 29901 f4b3f8fbf599
permissions -rw-r--r--
added Determinants to Library
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(*  Title:      HOL/Library/Multiset.thy
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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 Plain "~~/src/HOL/List"
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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 . f x > 0}}"
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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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definition Mempty :: "'a multiset"  ("{#}") where
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  [code del]: "{#} = Abs_multiset (\<lambda>a. 0)"
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definition single :: "'a => 'a multiset" where
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  [code del]: "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
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definition count :: "'a multiset => 'a => nat" where
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  "count = Rep_multiset"
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definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
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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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abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
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  "a :# M == 0 < count M a"
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notation (xsymbols)
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  Melem (infix "\<in>#" 50)
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syntax
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  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")
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translations
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  "{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
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definition set_of :: "'a multiset => 'a set" where
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  "set_of M = {x. x :# M}"
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instantiation multiset :: (type) "{plus, minus, zero, size}" 
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begin
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definition union_def [code del]:
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  "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
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definition diff_def [code del]:
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  "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
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definition Zero_multiset_def [simp]:
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  "0 = {#}"
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definition size_def:
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  "size M = setsum (count M) (set_of M)"
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instance ..
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end
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definition
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  multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
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  "multiset_inter A B = A - (A - B)"
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text {* Multiset Enumeration *}
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syntax
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  "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
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translations
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  "{#x, xs#}" == "{#x#} + {#xs#}"
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  "{#x#}" == "CONST single x"
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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: "(\<lambda>a. 0) \<in> multiset"
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by (simp add: multiset_def)
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lemma only1_in_multiset: "(\<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:
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  "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
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apply (simp add: multiset_def)
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apply (drule (1) finite_UnI)
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apply (simp del: finite_Un add: Un_def)
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done
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lemma diff_preserves_multiset:
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  "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
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apply (simp add: multiset_def)
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apply (rule finite_subset)
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 apply auto
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done
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lemma MCollect_preserves_multiset:
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  "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
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apply (simp add: multiset_def)
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apply (rule finite_subset, auto)
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done
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lemmas in_multiset = const0_in_multiset only1_in_multiset
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  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
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subsection {* Algebraic properties *}
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subsubsection {* Union *}
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lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
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by (simp add: union_def Mempty_def in_multiset)
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lemma union_commute: "M + N = N + (M::'a multiset)"
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by (simp add: union_def add_ac in_multiset)
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lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
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by (simp add: union_def add_ac in_multiset)
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lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
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proof -
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  have "M + (N + K) = (N + K) + M" by (rule union_commute)
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  also have "\<dots> = N + (K + M)" by (rule union_assoc)
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  also have "K + M = M + K" by (rule union_commute)
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  finally show ?thesis .
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qed
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lemmas union_ac = union_assoc union_commute union_lcomm
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instance multiset :: (type) comm_monoid_add
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proof
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  fix a b c :: "'a multiset"
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  show "(a + b) + c = a + (b + c)" by (rule union_assoc)
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  show "a + b = b + a" by (rule union_commute)
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  show "0 + a = a" by simp
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qed
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subsubsection {* Difference *}
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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by (simp add: Mempty_def diff_def in_multiset)
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lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
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by (simp add: union_def diff_def in_multiset)
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lemma diff_cancel: "A - A = {#}"
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by (simp add: diff_def Mempty_def)
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subsubsection {* Count of elements *}
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lemma count_empty [simp]: "count {#} a = 0"
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by (simp add: count_def Mempty_def in_multiset)
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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by (simp add: count_def single_def in_multiset)
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lemma count_union [simp]: "count (M + N) a = count M a + count N a"
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by (simp add: count_def union_def in_multiset)
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
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by (simp add: count_def diff_def in_multiset)
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lemma count_MCollect [simp]:
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  "count {# x:#M. P x #} a = (if P a then count M a else 0)"
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by (simp add: count_def MCollect_def in_multiset)
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subsubsection {* Set of elements *}
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lemma set_of_empty [simp]: "set_of {#} = {}"
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by (simp add: set_of_def)
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lemma set_of_single [simp]: "set_of {#b#} = {b}"
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by (simp add: set_of_def)
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lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
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by (auto simp add: set_of_def)
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lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
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by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
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lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
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by (auto simp add: set_of_def)
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lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
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by (auto simp add: set_of_def)
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subsubsection {* Size *}
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lemma size_empty [simp]: "size {#} = 0"
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by (simp add: size_def)
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lemma size_single [simp]: "size {#b#} = 1"
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by (simp add: size_def)
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lemma finite_set_of [iff]: "finite (set_of M)"
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using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
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lemma setsum_count_Int:
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  "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
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apply (induct rule: finite_induct)
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 apply simp
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apply (simp add: Int_insert_left set_of_def)
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done
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lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
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apply (unfold size_def)
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apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
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 prefer 2
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 apply (rule ext, simp)
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apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
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apply (subst Int_commute)
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apply (simp (no_asm_simp) add: setsum_count_Int)
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done
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lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
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apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
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apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
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done
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lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
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by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
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lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
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apply (unfold size_def)
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apply (drule setsum_SucD)
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apply auto
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done
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subsubsection {* Equality of multisets *}
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lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
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by (simp add: count_def expand_fun_eq)
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
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lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
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by (auto simp add: single_def in_multiset expand_fun_eq)
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lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
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by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
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lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
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by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
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lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
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by (simp add: union_def in_multiset expand_fun_eq)
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lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
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by (simp add: union_def in_multiset expand_fun_eq)
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lemma union_is_single:
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  "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
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apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
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apply blast
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done
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lemma single_is_union:
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  "({#a#} = M + N) \<longleftrightarrow> ({#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 in_multiset expand_fun_eq)
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apply (blast dest: sym)
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done
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lemma add_eq_conv_diff:
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  "(M + {#a#} = N + {#b#}) =
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   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
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using [[simproc del: neq]]
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apply (unfold single_def union_def diff_def)
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apply (simp (no_asm) add: in_multiset expand_fun_eq)
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apply (rule conjI, force, safe, simp_all)
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apply (simp add: eq_sym_conv)
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done
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declare Rep_multiset_inject [symmetric, simp del]
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instance multiset :: (type) cancel_ab_semigroup_add
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proof
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  fix a b c :: "'a multiset"
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  show "a + b = a + c \<Longrightarrow> b = c" by simp
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qed
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lemma insert_DiffM:
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  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
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by (clarsimp simp: multiset_eq_conv_count_eq)
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lemma insert_DiffM2[simp]:
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  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
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by (clarsimp simp: multiset_eq_conv_count_eq)
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lemma multi_union_self_other_eq: 
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  "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
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by (induct A arbitrary: X Y) auto
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lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
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by (metis single_not_empty union_empty union_left_cancel)
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lemma insert_noteq_member: 
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  assumes BC: "B + {#b#} = C + {#c#}"
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   and bnotc: "b \<noteq> c"
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  shows "c \<in># B"
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proof -
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  have "c \<in># C + {#c#}" by simp
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  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
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  then have "c \<in># B + {#b#}" using BC by simp
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  then show "c \<in># B" using nc by simp
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qed
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lemma add_eq_conv_ex:
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  "(M + {#a#} = N + {#b#}) =
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    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
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by (auto simp add: add_eq_conv_diff)
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lemma empty_multiset_count:
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  "(\<forall>x. count A x = 0) = (A = {#})"
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by (metis count_empty multiset_eq_conv_count_eq)
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kleing@15869
   332
subsubsection {* Intersection *}
kleing@15869
   333
kleing@15869
   334
lemma multiset_inter_count:
nipkow@26178
   335
  "count (A #\<inter> B) x = min (count A x) (count B x)"
nipkow@26178
   336
by (simp add: multiset_inter_def min_def)
kleing@15869
   337
kleing@15869
   338
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
nipkow@26178
   339
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214
   340
    min_max.inf_commute)
kleing@15869
   341
kleing@15869
   342
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
nipkow@26178
   343
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214
   344
    min_max.inf_assoc)
kleing@15869
   345
kleing@15869
   346
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
nipkow@26178
   347
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
kleing@15869
   348
wenzelm@17161
   349
lemmas multiset_inter_ac =
wenzelm@17161
   350
  multiset_inter_commute
wenzelm@17161
   351
  multiset_inter_assoc
wenzelm@17161
   352
  multiset_inter_left_commute
kleing@15869
   353
bulwahn@26143
   354
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
nipkow@26178
   355
by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
bulwahn@26143
   356
kleing@15869
   357
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
nipkow@26178
   358
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
wenzelm@17161
   359
    split: split_if_asm)
nipkow@26178
   360
apply clarsimp
nipkow@26178
   361
apply (erule_tac x = a in allE)
nipkow@26178
   362
apply auto
nipkow@26178
   363
done
kleing@15869
   364
wenzelm@10249
   365
nipkow@26016
   366
subsubsection {* Comprehension (filter) *}
nipkow@26016
   367
haftmann@28708
   368
lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
nipkow@26178
   369
by (simp add: MCollect_def Mempty_def Abs_multiset_inject
wenzelm@26145
   370
    in_multiset expand_fun_eq)
nipkow@26016
   371
haftmann@28708
   372
lemma MCollect_single [simp]:
nipkow@26178
   373
  "MCollect {#x#} P = (if P x then {#x#} else {#})"
nipkow@26178
   374
by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
wenzelm@26145
   375
    in_multiset expand_fun_eq)
nipkow@26016
   376
haftmann@28708
   377
lemma MCollect_union [simp]:
nipkow@26016
   378
  "MCollect (M+N) f = MCollect M f + MCollect N f"
nipkow@26178
   379
by (simp add: MCollect_def union_def Abs_multiset_inject
wenzelm@26145
   380
    in_multiset expand_fun_eq)
nipkow@26016
   381
nipkow@26016
   382
nipkow@26016
   383
subsection {* Induction and case splits *}
wenzelm@10249
   384
wenzelm@10249
   385
lemma setsum_decr:
wenzelm@11701
   386
  "finite F ==> (0::nat) < f a ==>
paulson@15072
   387
    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
nipkow@26178
   388
apply (induct rule: finite_induct)
nipkow@26178
   389
 apply auto
nipkow@26178
   390
apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@26178
   391
done
wenzelm@10249
   392
wenzelm@10313
   393
lemma rep_multiset_induct_aux:
nipkow@26178
   394
assumes 1: "P (\<lambda>a. (0::nat))"
nipkow@26178
   395
  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow@26178
   396
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow@26178
   397
apply (unfold multiset_def)
nipkow@26178
   398
apply (induct_tac n, simp, clarify)
nipkow@26178
   399
 apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow@26178
   400
  apply simp
nipkow@26178
   401
  apply (rule 1)
nipkow@26178
   402
 apply (rule ext, force, clarify)
nipkow@26178
   403
apply (frule setsum_SucD, clarify)
nipkow@26178
   404
apply (rename_tac a)
nipkow@26178
   405
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow@26178
   406
 prefer 2
nipkow@26178
   407
 apply (rule finite_subset)
nipkow@26178
   408
  prefer 2
nipkow@26178
   409
  apply assumption
nipkow@26178
   410
 apply simp
nipkow@26178
   411
 apply blast
nipkow@26178
   412
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow@26178
   413
 prefer 2
nipkow@26178
   414
 apply (rule ext)
nipkow@26178
   415
 apply (simp (no_asm_simp))
nipkow@26178
   416
 apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow@26178
   417
apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow@26178
   418
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow@26178
   419
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow@26178
   420
 prefer 2
nipkow@26178
   421
 apply blast
nipkow@26178
   422
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow@26178
   423
 prefer 2
nipkow@26178
   424
 apply blast
nipkow@26178
   425
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow@26178
   426
done
wenzelm@10249
   427
wenzelm@10313
   428
theorem rep_multiset_induct:
nipkow@11464
   429
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701
   430
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
nipkow@26178
   431
using rep_multiset_induct_aux by blast
wenzelm@10249
   432
wenzelm@18258
   433
theorem multiset_induct [case_names empty add, induct type: multiset]:
nipkow@26178
   434
assumes empty: "P {#}"
nipkow@26178
   435
  and add: "!!M x. P M ==> P (M + {#x#})"
nipkow@26178
   436
shows "P M"
wenzelm@10249
   437
proof -
wenzelm@10249
   438
  note defns = union_def single_def Mempty_def
wenzelm@10249
   439
  show ?thesis
wenzelm@10249
   440
    apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313
   441
    apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@18258
   442
     apply (rule empty [unfolded defns])
paulson@15072
   443
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249
   444
     prefer 2
wenzelm@10249
   445
     apply (simp add: expand_fun_eq)
wenzelm@10249
   446
    apply (erule ssubst)
wenzelm@17200
   447
    apply (erule Abs_multiset_inverse [THEN subst])
nipkow@26016
   448
    apply (drule add [unfolded defns, simplified])
nipkow@26016
   449
    apply(simp add:in_multiset)
wenzelm@10249
   450
    done
wenzelm@10249
   451
qed
wenzelm@10249
   452
kleing@25610
   453
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
nipkow@26178
   454
by (induct M) auto
kleing@25610
   455
kleing@25610
   456
lemma multiset_cases [cases type, case_names empty add]:
nipkow@26178
   457
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow@26178
   458
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow@26178
   459
shows "P"
kleing@25610
   460
proof (cases "M = {#}")
wenzelm@26145
   461
  assume "M = {#}" then show ?thesis using em by simp
kleing@25610
   462
next
kleing@25610
   463
  assume "M \<noteq> {#}"
kleing@25610
   464
  then obtain M' m where "M = M' + {#m#}" 
kleing@25610
   465
    by (blast dest: multi_nonempty_split)
wenzelm@26145
   466
  then show ?thesis using add by simp
kleing@25610
   467
qed
kleing@25610
   468
kleing@25610
   469
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
nipkow@26178
   470
apply (cases M)
nipkow@26178
   471
 apply simp
nipkow@26178
   472
apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow@26178
   473
done
kleing@25610
   474
nipkow@26033
   475
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
nipkow@26178
   476
apply (subst multiset_eq_conv_count_eq)
nipkow@26178
   477
apply auto
nipkow@26178
   478
done
wenzelm@10249
   479
kleing@15869
   480
declare multiset_typedef [simp del]
wenzelm@10249
   481
kleing@25610
   482
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
nipkow@26178
   483
by (cases "B = {#}") (auto dest: multi_member_split)
wenzelm@26145
   484
wenzelm@17161
   485
nipkow@26016
   486
subsection {* Orderings *}
wenzelm@10249
   487
wenzelm@10249
   488
subsubsection {* Well-foundedness *}
wenzelm@10249
   489
haftmann@28708
   490
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
haftmann@28708
   491
  [code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
berghofe@23751
   492
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249
   493
haftmann@28708
   494
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
berghofe@23751
   495
  "mult r = (mult1 r)\<^sup>+"
wenzelm@10249
   496
berghofe@23751
   497
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
nipkow@26178
   498
by (simp add: mult1_def)
wenzelm@10249
   499
berghofe@23751
   500
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
berghofe@23751
   501
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
berghofe@23751
   502
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@19582
   503
  (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
wenzelm@10249
   504
proof (unfold mult1_def)
berghofe@23751
   505
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   506
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
berghofe@23751
   507
  let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249
   508
berghofe@23751
   509
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
wenzelm@18258
   510
  then have "\<exists>a' M0' K.
nipkow@11464
   511
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
wenzelm@18258
   512
  then show "?case1 \<or> ?case2"
wenzelm@10249
   513
  proof (elim exE conjE)
wenzelm@10249
   514
    fix a' M0' K
wenzelm@10249
   515
    assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249
   516
    assume "M0 + {#a#} = M0' + {#a'#}"
wenzelm@18258
   517
    then have "M0 = M0' \<and> a = a' \<or>
nipkow@11464
   518
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249
   519
      by (simp only: add_eq_conv_ex)
wenzelm@18258
   520
    then show ?thesis
wenzelm@10249
   521
    proof (elim disjE conjE exE)
wenzelm@10249
   522
      assume "M0 = M0'" "a = a'"
nipkow@11464
   523
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@18258
   524
      then have ?case2 .. then show ?thesis ..
wenzelm@10249
   525
    next
wenzelm@10249
   526
      fix K'
wenzelm@10249
   527
      assume "M0' = K' + {#a#}"
wenzelm@10249
   528
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249
   529
wenzelm@10249
   530
      assume "M0 = K' + {#a'#}"
wenzelm@10249
   531
      with r have "?R (K' + K) M0" by blast
wenzelm@18258
   532
      with n have ?case1 by simp then show ?thesis ..
wenzelm@10249
   533
    qed
wenzelm@10249
   534
  qed
wenzelm@10249
   535
qed
wenzelm@10249
   536
berghofe@23751
   537
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249
   538
proof
wenzelm@10249
   539
  let ?R = "mult1 r"
wenzelm@10249
   540
  let ?W = "acc ?R"
wenzelm@10249
   541
  {
wenzelm@10249
   542
    fix M M0 a
berghofe@23751
   543
    assume M0: "M0 \<in> ?W"
berghofe@23751
   544
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751
   545
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
berghofe@23751
   546
    have "M0 + {#a#} \<in> ?W"
berghofe@23751
   547
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
   548
      fix N
berghofe@23751
   549
      assume "(N, M0 + {#a#}) \<in> ?R"
berghofe@23751
   550
      then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
berghofe@23751
   551
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249
   552
        by (rule less_add)
berghofe@23751
   553
      then show "N \<in> ?W"
wenzelm@10249
   554
      proof (elim exE disjE conjE)
berghofe@23751
   555
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
berghofe@23751
   556
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
berghofe@23751
   557
        from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
berghofe@23751
   558
        then show "N \<in> ?W" by (simp only: N)
wenzelm@10249
   559
      next
wenzelm@10249
   560
        fix K
wenzelm@10249
   561
        assume N: "N = M0 + K"
berghofe@23751
   562
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
berghofe@23751
   563
        then have "M0 + K \<in> ?W"
wenzelm@10249
   564
        proof (induct K)
wenzelm@18730
   565
          case empty
berghofe@23751
   566
          from M0 show "M0 + {#} \<in> ?W" by simp
wenzelm@18730
   567
        next
wenzelm@18730
   568
          case (add K x)
berghofe@23751
   569
          from add.prems have "(x, a) \<in> r" by simp
berghofe@23751
   570
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
berghofe@23751
   571
          moreover from add have "M0 + K \<in> ?W" by simp
berghofe@23751
   572
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
berghofe@23751
   573
          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249
   574
        qed
berghofe@23751
   575
        then show "N \<in> ?W" by (simp only: N)
wenzelm@10249
   576
      qed
wenzelm@10249
   577
    qed
wenzelm@10249
   578
  } note tedious_reasoning = this
wenzelm@10249
   579
berghofe@23751
   580
  assume wf: "wf r"
wenzelm@10249
   581
  fix M
berghofe@23751
   582
  show "M \<in> ?W"
wenzelm@10249
   583
  proof (induct M)
berghofe@23751
   584
    show "{#} \<in> ?W"
wenzelm@10249
   585
    proof (rule accI)
berghofe@23751
   586
      fix b assume "(b, {#}) \<in> ?R"
berghofe@23751
   587
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
   588
    qed
wenzelm@10249
   589
berghofe@23751
   590
    fix M a assume "M \<in> ?W"
berghofe@23751
   591
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   592
    proof induct
wenzelm@10249
   593
      fix a
berghofe@23751
   594
      assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751
   595
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   596
      proof
berghofe@23751
   597
        fix M assume "M \<in> ?W"
berghofe@23751
   598
        then show "M + {#a#} \<in> ?W"
wenzelm@23373
   599
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
wenzelm@10249
   600
      qed
wenzelm@10249
   601
    qed
berghofe@23751
   602
    from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
wenzelm@10249
   603
  qed
wenzelm@10249
   604
qed
wenzelm@10249
   605
berghofe@23751
   606
theorem wf_mult1: "wf r ==> wf (mult1 r)"
nipkow@26178
   607
by (rule acc_wfI) (rule all_accessible)
wenzelm@10249
   608
berghofe@23751
   609
theorem wf_mult: "wf r ==> wf (mult r)"
nipkow@26178
   610
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249
   611
wenzelm@10249
   612
wenzelm@10249
   613
subsubsection {* Closure-free presentation *}
wenzelm@10249
   614
wenzelm@10249
   615
(*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249
   616
wenzelm@10249
   617
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
nipkow@26178
   618
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   619
wenzelm@10249
   620
text {* One direction. *}
wenzelm@10249
   621
wenzelm@10249
   622
lemma mult_implies_one_step:
berghofe@23751
   623
  "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464
   624
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
berghofe@23751
   625
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
nipkow@26178
   626
apply (unfold mult_def mult1_def set_of_def)
nipkow@26178
   627
apply (erule converse_trancl_induct, clarify)
nipkow@26178
   628
 apply (rule_tac x = M0 in exI, simp, clarify)
nipkow@26178
   629
apply (case_tac "a :# K")
nipkow@26178
   630
 apply (rule_tac x = I in exI)
nipkow@26178
   631
 apply (simp (no_asm))
nipkow@26178
   632
 apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
nipkow@26178
   633
 apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@26178
   634
 apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow@26178
   635
 apply (simp add: diff_union_single_conv)
nipkow@26178
   636
 apply (simp (no_asm_use) add: trans_def)
nipkow@26178
   637
 apply blast
nipkow@26178
   638
apply (subgoal_tac "a :# I")
nipkow@26178
   639
 apply (rule_tac x = "I - {#a#}" in exI)
nipkow@26178
   640
 apply (rule_tac x = "J + {#a#}" in exI)
nipkow@26178
   641
 apply (rule_tac x = "K + Ka" in exI)
nipkow@26178
   642
 apply (rule conjI)
nipkow@26178
   643
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow@26178
   644
 apply (rule conjI)
nipkow@26178
   645
  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow@26178
   646
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow@26178
   647
 apply (simp (no_asm_use) add: trans_def)
nipkow@26178
   648
 apply blast
nipkow@26178
   649
apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow@26178
   650
 apply simp
nipkow@26178
   651
apply (simp (no_asm))
nipkow@26178
   652
done
wenzelm@10249
   653
wenzelm@10249
   654
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
nipkow@26178
   655
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   656
nipkow@11464
   657
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
nipkow@26178
   658
apply (erule size_eq_Suc_imp_elem [THEN exE])
nipkow@26178
   659
apply (drule elem_imp_eq_diff_union, auto)
nipkow@26178
   660
done
wenzelm@10249
   661
wenzelm@10249
   662
lemma one_step_implies_mult_aux:
berghofe@23751
   663
  "trans r ==>
berghofe@23751
   664
    \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
berghofe@23751
   665
      --> (I + K, I + J) \<in> mult r"
nipkow@26178
   666
apply (induct_tac n, auto)
nipkow@26178
   667
apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow@26178
   668
apply (rename_tac "J'", simp)
nipkow@26178
   669
apply (erule notE, auto)
nipkow@26178
   670
apply (case_tac "J' = {#}")
nipkow@26178
   671
 apply (simp add: mult_def)
nipkow@26178
   672
 apply (rule r_into_trancl)
nipkow@26178
   673
 apply (simp add: mult1_def set_of_def, blast)
nipkow@26178
   674
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@26178
   675
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@26178
   676
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
nipkow@26178
   677
apply (erule ssubst)
nipkow@26178
   678
apply (simp add: Ball_def, auto)
nipkow@26178
   679
apply (subgoal_tac
nipkow@26178
   680
  "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
nipkow@26178
   681
    (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow@26178
   682
 prefer 2
nipkow@26178
   683
 apply force
nipkow@26178
   684
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
nipkow@26178
   685
apply (erule trancl_trans)
nipkow@26178
   686
apply (rule r_into_trancl)
nipkow@26178
   687
apply (simp add: mult1_def set_of_def)
nipkow@26178
   688
apply (rule_tac x = a in exI)
nipkow@26178
   689
apply (rule_tac x = "I + J'" in exI)
nipkow@26178
   690
apply (simp add: union_ac)
nipkow@26178
   691
done
wenzelm@10249
   692
wenzelm@17161
   693
lemma one_step_implies_mult:
berghofe@23751
   694
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
berghofe@23751
   695
    ==> (I + K, I + J) \<in> mult r"
nipkow@26178
   696
using one_step_implies_mult_aux by blast
wenzelm@10249
   697
wenzelm@10249
   698
wenzelm@10249
   699
subsubsection {* Partial-order properties *}
wenzelm@10249
   700
haftmann@26567
   701
instantiation multiset :: (order) order
haftmann@26567
   702
begin
wenzelm@10249
   703
haftmann@28708
   704
definition less_multiset_def [code del]:
haftmann@28708
   705
  "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
haftmann@26567
   706
haftmann@28708
   707
definition le_multiset_def [code del]:
haftmann@28708
   708
  "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249
   709
berghofe@23751
   710
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
nipkow@26178
   711
unfolding trans_def by (blast intro: order_less_trans)
wenzelm@10249
   712
wenzelm@10249
   713
text {*
wenzelm@10249
   714
 \medskip Irreflexivity.
wenzelm@10249
   715
*}
wenzelm@10249
   716
wenzelm@10249
   717
lemma mult_irrefl_aux:
nipkow@26178
   718
  "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
nipkow@26178
   719
by (induct rule: finite_induct) (auto intro: order_less_trans)
wenzelm@10249
   720
wenzelm@17161
   721
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
nipkow@26178
   722
apply (unfold less_multiset_def, auto)
nipkow@26178
   723
apply (drule trans_base_order [THEN mult_implies_one_step], auto)
nipkow@26178
   724
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
nipkow@26178
   725
apply (simp add: set_of_eq_empty_iff)
nipkow@26178
   726
done
wenzelm@10249
   727
wenzelm@10249
   728
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
nipkow@26178
   729
using insert mult_less_not_refl by fast
wenzelm@10249
   730
wenzelm@10249
   731
wenzelm@10249
   732
text {* Transitivity. *}
wenzelm@10249
   733
wenzelm@10249
   734
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
nipkow@26178
   735
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@10249
   736
wenzelm@10249
   737
text {* Asymmetry. *}
wenzelm@10249
   738
nipkow@11464
   739
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
nipkow@26178
   740
apply auto
nipkow@26178
   741
apply (rule mult_less_not_refl [THEN notE])
nipkow@26178
   742
apply (erule mult_less_trans, assumption)
nipkow@26178
   743
done
wenzelm@10249
   744
wenzelm@10249
   745
theorem mult_less_asym:
nipkow@26178
   746
  "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
nipkow@26178
   747
using mult_less_not_sym by blast
wenzelm@10249
   748
wenzelm@10249
   749
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
nipkow@26178
   750
unfolding le_multiset_def by auto
wenzelm@10249
   751
wenzelm@10249
   752
text {* Anti-symmetry. *}
wenzelm@10249
   753
wenzelm@10249
   754
theorem mult_le_antisym:
nipkow@26178
   755
  "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
nipkow@26178
   756
unfolding le_multiset_def by (blast dest: mult_less_not_sym)
wenzelm@10249
   757
wenzelm@10249
   758
text {* Transitivity. *}
wenzelm@10249
   759
wenzelm@10249
   760
theorem mult_le_trans:
nipkow@26178
   761
  "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
nipkow@26178
   762
unfolding le_multiset_def by (blast intro: mult_less_trans)
wenzelm@10249
   763
wenzelm@11655
   764
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
nipkow@26178
   765
unfolding le_multiset_def by auto
wenzelm@10249
   766
haftmann@27682
   767
instance proof
haftmann@27682
   768
qed (auto simp add: mult_less_le dest: mult_le_antisym elim: mult_le_trans)
wenzelm@10277
   769
haftmann@26567
   770
end
haftmann@26567
   771
wenzelm@10249
   772
wenzelm@10249
   773
subsubsection {* Monotonicity of multiset union *}
wenzelm@10249
   774
wenzelm@17161
   775
lemma mult1_union:
nipkow@26178
   776
  "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
nipkow@26178
   777
apply (unfold mult1_def)
nipkow@26178
   778
apply auto
nipkow@26178
   779
apply (rule_tac x = a in exI)
nipkow@26178
   780
apply (rule_tac x = "C + M0" in exI)
nipkow@26178
   781
apply (simp add: union_assoc)
nipkow@26178
   782
done
wenzelm@10249
   783
wenzelm@10249
   784
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
nipkow@26178
   785
apply (unfold less_multiset_def mult_def)
nipkow@26178
   786
apply (erule trancl_induct)
nipkow@26178
   787
 apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
nipkow@26178
   788
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
nipkow@26178
   789
done
wenzelm@10249
   790
wenzelm@10249
   791
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
nipkow@26178
   792
apply (subst union_commute [of B C])
nipkow@26178
   793
apply (subst union_commute [of D C])
nipkow@26178
   794
apply (erule union_less_mono2)
nipkow@26178
   795
done
wenzelm@10249
   796
wenzelm@17161
   797
lemma union_less_mono:
nipkow@26178
   798
  "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
nipkow@26178
   799
by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249
   800
wenzelm@17161
   801
lemma union_le_mono:
nipkow@26178
   802
  "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
nipkow@26178
   803
unfolding le_multiset_def
nipkow@26178
   804
by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249
   805
wenzelm@17161
   806
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
nipkow@26178
   807
apply (unfold le_multiset_def less_multiset_def)
nipkow@26178
   808
apply (case_tac "M = {#}")
nipkow@26178
   809
 prefer 2
nipkow@26178
   810
 apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
nipkow@26178
   811
  prefer 2
nipkow@26178
   812
  apply (rule one_step_implies_mult)
nipkow@26178
   813
    apply (simp only: trans_def)
nipkow@26178
   814
    apply auto
nipkow@26178
   815
done
wenzelm@10249
   816
wenzelm@17161
   817
lemma union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072
   818
proof -
wenzelm@17200
   819
  have "A + {#} <= A + B" by (blast intro: union_le_mono)
wenzelm@18258
   820
  then show ?thesis by simp
paulson@15072
   821
qed
paulson@15072
   822
wenzelm@17161
   823
lemma union_upper2: "B <= A + (B::'a::order multiset)"
nipkow@26178
   824
by (subst union_commute) (rule union_upper1)
paulson@15072
   825
nipkow@23611
   826
instance multiset :: (order) pordered_ab_semigroup_add
nipkow@26178
   827
apply intro_classes
nipkow@26178
   828
apply (erule union_le_mono[OF mult_le_refl])
nipkow@26178
   829
done
wenzelm@26145
   830
paulson@15072
   831
wenzelm@17200
   832
subsection {* Link with lists *}
paulson@15072
   833
nipkow@26016
   834
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
wenzelm@26145
   835
  "multiset_of [] = {#}" |
wenzelm@26145
   836
  "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072
   837
paulson@15072
   838
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
nipkow@26178
   839
by (induct x) auto
paulson@15072
   840
paulson@15072
   841
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
nipkow@26178
   842
by (induct x) auto
paulson@15072
   843
paulson@15072
   844
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
nipkow@26178
   845
by (induct x) auto
kleing@15867
   846
kleing@15867
   847
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
nipkow@26178
   848
by (induct xs) auto
paulson@15072
   849
wenzelm@18258
   850
lemma multiset_of_append [simp]:
nipkow@26178
   851
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
nipkow@26178
   852
by (induct xs arbitrary: ys) (auto simp: union_ac)
wenzelm@18730
   853
paulson@15072
   854
lemma surj_multiset_of: "surj multiset_of"
nipkow@26178
   855
apply (unfold surj_def)
nipkow@26178
   856
apply (rule allI)
nipkow@26178
   857
apply (rule_tac M = y in multiset_induct)
nipkow@26178
   858
 apply auto
nipkow@26178
   859
apply (rule_tac x = "x # xa" in exI)
nipkow@26178
   860
apply auto
nipkow@26178
   861
done
wenzelm@10249
   862
nipkow@25162
   863
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
nipkow@26178
   864
by (induct x) auto
paulson@15072
   865
wenzelm@17200
   866
lemma distinct_count_atmost_1:
nipkow@26178
   867
  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
nipkow@26178
   868
apply (induct x, simp, rule iffI, simp_all)
nipkow@26178
   869
apply (rule conjI)
nipkow@26178
   870
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
nipkow@26178
   871
apply (erule_tac x = a in allE, simp, clarify)
nipkow@26178
   872
apply (erule_tac x = aa in allE, simp)
nipkow@26178
   873
done
paulson@15072
   874
wenzelm@17200
   875
lemma multiset_of_eq_setD:
kleing@15867
   876
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
nipkow@26178
   877
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
kleing@15867
   878
wenzelm@17200
   879
lemma set_eq_iff_multiset_of_eq_distinct:
wenzelm@26145
   880
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
wenzelm@26145
   881
    (set x = set y) = (multiset_of x = multiset_of y)"
nipkow@26178
   882
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
paulson@15072
   883
wenzelm@17200
   884
lemma set_eq_iff_multiset_of_remdups_eq:
paulson@15072
   885
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
nipkow@26178
   886
apply (rule iffI)
nipkow@26178
   887
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
nipkow@26178
   888
apply (drule distinct_remdups [THEN distinct_remdups
wenzelm@26145
   889
      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
nipkow@26178
   890
apply simp
nipkow@26178
   891
done
wenzelm@10249
   892
wenzelm@18258
   893
lemma multiset_of_compl_union [simp]:
nipkow@26178
   894
  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
nipkow@26178
   895
by (induct xs) (auto simp: union_ac)
paulson@15072
   896
wenzelm@17200
   897
lemma count_filter:
nipkow@26178
   898
  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
nipkow@26178
   899
by (induct xs) auto
kleing@15867
   900
bulwahn@26143
   901
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
nipkow@26178
   902
apply (induct ls arbitrary: i)
nipkow@26178
   903
 apply simp
nipkow@26178
   904
apply (case_tac i)
nipkow@26178
   905
 apply auto
nipkow@26178
   906
done
bulwahn@26143
   907
bulwahn@26143
   908
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
nipkow@26178
   909
by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
bulwahn@26143
   910
bulwahn@26143
   911
lemma multiset_of_eq_length:
nipkow@26178
   912
assumes "multiset_of xs = multiset_of ys"
nipkow@26178
   913
shows "length xs = length ys"
nipkow@26178
   914
using assms
bulwahn@26143
   915
proof (induct arbitrary: ys rule: length_induct)
bulwahn@26143
   916
  case (1 xs ys)
bulwahn@26143
   917
  show ?case
bulwahn@26143
   918
  proof (cases xs)
wenzelm@26145
   919
    case Nil with "1.prems" show ?thesis by simp
bulwahn@26143
   920
  next
bulwahn@26143
   921
    case (Cons x xs')
bulwahn@26143
   922
    note xCons = Cons
bulwahn@26143
   923
    show ?thesis
bulwahn@26143
   924
    proof (cases ys)
bulwahn@26143
   925
      case Nil
wenzelm@26145
   926
      with "1.prems" Cons show ?thesis by simp
bulwahn@26143
   927
    next
bulwahn@26143
   928
      case (Cons y ys')
bulwahn@26143
   929
      have x_in_ys: "x = y \<or> x \<in> set ys'"
bulwahn@26143
   930
      proof (cases "x = y")
wenzelm@26145
   931
	case True then show ?thesis ..
bulwahn@26143
   932
      next
bulwahn@26143
   933
	case False
wenzelm@26145
   934
	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
bulwahn@26143
   935
	with False show ?thesis by (simp add: mem_set_multiset_eq)
bulwahn@26143
   936
      qed
wenzelm@26145
   937
      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
wenzelm@26145
   938
	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
wenzelm@26145
   939
      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
bulwahn@26143
   940
	apply -
bulwahn@26143
   941
	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
bulwahn@26143
   942
	apply fastsimp
bulwahn@26143
   943
	done
wenzelm@26145
   944
      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
wenzelm@26145
   945
      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
bulwahn@26143
   946
      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
bulwahn@26143
   947
    qed
bulwahn@26143
   948
  qed
bulwahn@26143
   949
qed
bulwahn@26143
   950
wenzelm@26145
   951
text {*
wenzelm@26145
   952
  This lemma shows which properties suffice to show that a function
wenzelm@26145
   953
  @{text "f"} with @{text "f xs = ys"} behaves like sort.
wenzelm@26145
   954
*}
wenzelm@26145
   955
lemma properties_for_sort:
wenzelm@26145
   956
  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
bulwahn@26143
   957
proof (induct xs arbitrary: ys)
wenzelm@26145
   958
  case Nil then show ?case by simp
bulwahn@26143
   959
next
bulwahn@26143
   960
  case (Cons x xs)
wenzelm@26145
   961
  then have "x \<in> set ys"
wenzelm@26145
   962
    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
bulwahn@26143
   963
  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
bulwahn@26143
   964
    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
bulwahn@26143
   965
qed
bulwahn@26143
   966
kleing@15867
   967
paulson@15072
   968
subsection {* Pointwise ordering induced by count *}
paulson@15072
   969
haftmann@28708
   970
definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
haftmann@28708
   971
  [code del]: "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
wenzelm@26145
   972
haftmann@28708
   973
definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
haftmann@28708
   974
  [code del]: "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
kleing@25610
   975
wenzelm@26145
   976
notation mset_le  (infix "\<subseteq>#" 50)
wenzelm@26145
   977
notation mset_less  (infix "\<subset>#" 50)
paulson@15072
   978
nipkow@23611
   979
lemma mset_le_refl[simp]: "A \<le># A"
nipkow@26178
   980
unfolding mset_le_def by auto
paulson@15072
   981
wenzelm@26145
   982
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
nipkow@26178
   983
unfolding mset_le_def by (fast intro: order_trans)
paulson@15072
   984
wenzelm@26145
   985
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
nipkow@26178
   986
apply (unfold mset_le_def)
nipkow@26178
   987
apply (rule multiset_eq_conv_count_eq [THEN iffD2])
nipkow@26178
   988
apply (blast intro: order_antisym)
nipkow@26178
   989
done
paulson@15072
   990
wenzelm@26145
   991
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
nipkow@26178
   992
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
nipkow@26178
   993
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
nipkow@26178
   994
done
paulson@15072
   995
nipkow@23611
   996
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
nipkow@26178
   997
unfolding mset_le_def by auto
paulson@15072
   998
nipkow@23611
   999
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
nipkow@26178
  1000
unfolding mset_le_def by auto
paulson@15072
  1001
nipkow@23611
  1002
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
nipkow@26178
  1003
apply (unfold mset_le_def)
nipkow@26178
  1004
apply auto
nipkow@26178
  1005
apply (erule_tac x = a in allE)+
nipkow@26178
  1006
apply auto
nipkow@26178
  1007
done
paulson@15072
  1008
nipkow@23611
  1009
lemma mset_le_add_left[simp]: "A \<le># A + B"
nipkow@26178
  1010
unfolding mset_le_def by auto
paulson@15072
  1011
nipkow@23611
  1012
lemma mset_le_add_right[simp]: "B \<le># A + B"
nipkow@26178
  1013
unfolding mset_le_def by auto
paulson@15072
  1014
bulwahn@26143
  1015
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
nipkow@26178
  1016
by (simp add: mset_le_def)
bulwahn@26143
  1017
bulwahn@26143
  1018
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
nipkow@26178
  1019
by (simp add: multiset_eq_conv_count_eq mset_le_def)
bulwahn@26143
  1020
bulwahn@26143
  1021
lemma mset_le_multiset_union_diff_commute:
nipkow@26178
  1022
assumes "B \<le># A"
nipkow@26178
  1023
shows "A - B + C = A + C - B"
bulwahn@26143
  1024
proof -
wenzelm@26145
  1025
  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
wenzelm@26145
  1026
  from this obtain D where "A = B + D" ..
wenzelm@26145
  1027
  then show ?thesis
wenzelm@26145
  1028
    apply simp
wenzelm@26145
  1029
    apply (subst union_commute)
wenzelm@26145
  1030
    apply (subst multiset_diff_union_assoc)
wenzelm@26145
  1031
    apply simp
wenzelm@26145
  1032
    apply (simp add: diff_cancel)
wenzelm@26145
  1033
    apply (subst union_assoc)
wenzelm@26145
  1034
    apply (subst union_commute[of "B" _])
wenzelm@26145
  1035
    apply (subst multiset_diff_union_assoc)
wenzelm@26145
  1036
    apply simp
wenzelm@26145
  1037
    apply (simp add: diff_cancel)
wenzelm@26145
  1038
    done
bulwahn@26143
  1039
qed
bulwahn@26143
  1040
nipkow@23611
  1041
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
nipkow@26178
  1042
apply (induct xs)
nipkow@26178
  1043
 apply auto
nipkow@26178
  1044
apply (rule mset_le_trans)
nipkow@26178
  1045
 apply auto
nipkow@26178
  1046
done
nipkow@23611
  1047
wenzelm@26145
  1048
lemma multiset_of_update:
wenzelm@26145
  1049
  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
bulwahn@26143
  1050
proof (induct ls arbitrary: i)
wenzelm@26145
  1051
  case Nil then show ?case by simp
bulwahn@26143
  1052
next
bulwahn@26143
  1053
  case (Cons x xs)
bulwahn@26143
  1054
  show ?case
wenzelm@26145
  1055
  proof (cases i)
wenzelm@26145
  1056
    case 0 then show ?thesis by simp
wenzelm@26145
  1057
  next
wenzelm@26145
  1058
    case (Suc i')
wenzelm@26145
  1059
    with Cons show ?thesis
wenzelm@26145
  1060
      apply simp
wenzelm@26145
  1061
      apply (subst union_assoc)
wenzelm@26145
  1062
      apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
wenzelm@26145
  1063
      apply (subst union_assoc [symmetric])
wenzelm@26145
  1064
      apply simp
wenzelm@26145
  1065
      apply (rule mset_le_multiset_union_diff_commute)
wenzelm@26145
  1066
      apply (simp add: mset_le_single nth_mem_multiset_of)
wenzelm@26145
  1067
      done
bulwahn@26143
  1068
  qed
bulwahn@26143
  1069
qed
bulwahn@26143
  1070
wenzelm@26145
  1071
lemma multiset_of_swap:
wenzelm@26145
  1072
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
wenzelm@26145
  1073
    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
nipkow@26178
  1074
apply (case_tac "i = j")
nipkow@26178
  1075
 apply simp
nipkow@26178
  1076
apply (simp add: multiset_of_update)
nipkow@26178
  1077
apply (subst elem_imp_eq_diff_union[symmetric])
nipkow@26178
  1078
 apply (simp add: nth_mem_multiset_of)
nipkow@26178
  1079
apply simp
nipkow@26178
  1080
done
bulwahn@26143
  1081
haftmann@29509
  1082
interpretation mset_order!: order "op \<le>#" "op <#"
haftmann@27682
  1083
proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
haftmann@27682
  1084
  mset_le_trans simp: mset_less_def)
nipkow@23611
  1085
haftmann@29509
  1086
interpretation mset_order_cancel_semigroup!:
haftmann@29509
  1087
  pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
haftmann@27682
  1088
proof qed (erule mset_le_mono_add [OF mset_le_refl])
nipkow@23611
  1089
haftmann@29509
  1090
interpretation mset_order_semigroup_cancel!:
haftmann@29509
  1091
  pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
haftmann@27682
  1092
proof qed simp
paulson@15072
  1093
kleing@25610
  1094
wenzelm@26145
  1095
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
nipkow@26178
  1096
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178
  1097
apply (erule_tac x=x in allE)
nipkow@26178
  1098
apply auto
nipkow@26178
  1099
done
kleing@25610
  1100
wenzelm@26145
  1101
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
nipkow@26178
  1102
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178
  1103
apply (erule_tac x = x in allE)
nipkow@26178
  1104
apply auto
nipkow@26178
  1105
done
kleing@25610
  1106
  
wenzelm@26145
  1107
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
nipkow@26178
  1108
apply (rule conjI)
nipkow@26178
  1109
 apply (simp add: mset_lessD)
nipkow@26178
  1110
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178
  1111
apply safe
nipkow@26178
  1112
 apply (erule_tac x = a in allE)
nipkow@26178
  1113
 apply (auto split: split_if_asm)
nipkow@26178
  1114
done
kleing@25610
  1115
wenzelm@26145
  1116
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
nipkow@26178
  1117
apply (rule conjI)
nipkow@26178
  1118
 apply (simp add: mset_leD)
nipkow@26178
  1119
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
nipkow@26178
  1120
done
kleing@25610
  1121
kleing@25610
  1122
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" 
nipkow@26178
  1123
by (induct A) (auto simp: mset_le_def mset_less_def)
kleing@25610
  1124
kleing@25610
  1125
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
nipkow@26178
  1126
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1127
kleing@25610
  1128
lemma multi_psub_self[simp]: "A \<subset># A = False"
nipkow@26178
  1129
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1130
kleing@25610
  1131
lemma mset_less_add_bothsides:
kleing@25610
  1132
  "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
nipkow@26178
  1133
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1134
kleing@25610
  1135
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
nipkow@26178
  1136
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1137
kleing@25610
  1138
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
kleing@25610
  1139
proof (induct A arbitrary: B)
kleing@25610
  1140
  case (empty M)
wenzelm@26145
  1141
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
kleing@25610
  1142
  then obtain M' x where "M = M' + {#x#}" 
kleing@25610
  1143
    by (blast dest: multi_nonempty_split)
wenzelm@26145
  1144
  then show ?case by simp
kleing@25610
  1145
next
kleing@25610
  1146
  case (add S x T)
kleing@25610
  1147
  have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
kleing@25610
  1148
  have SxsubT: "S + {#x#} \<subset># T" by fact
wenzelm@26145
  1149
  then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
kleing@25610
  1150
  then obtain T' where T: "T = T' + {#x#}" 
kleing@25610
  1151
    by (blast dest: multi_member_split)
wenzelm@26145
  1152
  then have "S \<subset># T'" using SxsubT 
kleing@25610
  1153
    by (blast intro: mset_less_add_bothsides)
wenzelm@26145
  1154
  then have "size S < size T'" using IH by simp
wenzelm@26145
  1155
  then show ?case using T by simp
kleing@25610
  1156
qed
kleing@25610
  1157
haftmann@29509
  1158
lemmas mset_less_trans = mset_order.less_trans
kleing@25610
  1159
kleing@25610
  1160
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
nipkow@26178
  1161
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
kleing@25610
  1162
wenzelm@26145
  1163
kleing@25610
  1164
subsection {* Strong induction and subset induction for multisets *}
kleing@25610
  1165
nipkow@26016
  1166
text {* Well-foundedness of proper subset operator: *}
kleing@25610
  1167
wenzelm@26145
  1168
text {* proper multiset subset *}
kleing@25610
  1169
definition
wenzelm@26145
  1170
  mset_less_rel :: "('a multiset * 'a multiset) set" where
wenzelm@26145
  1171
  "mset_less_rel = {(A,B). A \<subset># B}"
kleing@25610
  1172
kleing@25610
  1173
lemma multiset_add_sub_el_shuffle: 
wenzelm@26145
  1174
  assumes "c \<in># B" and "b \<noteq> c" 
kleing@25610
  1175
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
kleing@25610
  1176
proof -
wenzelm@26145
  1177
  from `c \<in># B` obtain A where B: "B = A + {#c#}" 
kleing@25610
  1178
    by (blast dest: multi_member_split)
kleing@25610
  1179
  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
wenzelm@26145
  1180
  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
kleing@25610
  1181
    by (simp add: union_ac)
wenzelm@26145
  1182
  then show ?thesis using B by simp
kleing@25610
  1183
qed
kleing@25610
  1184
kleing@25610
  1185
lemma wf_mset_less_rel: "wf mset_less_rel"
nipkow@26178
  1186
apply (unfold mset_less_rel_def)
nipkow@26178
  1187
apply (rule wf_measure [THEN wf_subset, where f1=size])
nipkow@26178
  1188
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
nipkow@26178
  1189
done
kleing@25610
  1190
nipkow@26016
  1191
text {* The induction rules: *}
kleing@25610
  1192
kleing@25610
  1193
lemma full_multiset_induct [case_names less]:
nipkow@26178
  1194
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
nipkow@26178
  1195
shows "P B"
nipkow@26178
  1196
apply (rule wf_mset_less_rel [THEN wf_induct])
nipkow@26178
  1197
apply (rule ih, auto simp: mset_less_rel_def)
nipkow@26178
  1198
done
kleing@25610
  1199
kleing@25610
  1200
lemma multi_subset_induct [consumes 2, case_names empty add]:
nipkow@26178
  1201
assumes "F \<subseteq># A"
nipkow@26178
  1202
  and empty: "P {#}"
nipkow@26178
  1203
  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
nipkow@26178
  1204
shows "P F"
kleing@25610
  1205
proof -
kleing@25610
  1206
  from `F \<subseteq># A`
kleing@25610
  1207
  show ?thesis
kleing@25610
  1208
  proof (induct F)
kleing@25610
  1209
    show "P {#}" by fact
kleing@25610
  1210
  next
kleing@25610
  1211
    fix x F
kleing@25610
  1212
    assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
kleing@25610
  1213
    show "P (F + {#x#})"
kleing@25610
  1214
    proof (rule insert)
kleing@25610
  1215
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
wenzelm@26145
  1216
      from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
kleing@25610
  1217
      with P show "P F" .
kleing@25610
  1218
    qed
kleing@25610
  1219
  qed
kleing@25610
  1220
qed 
kleing@25610
  1221
nipkow@26016
  1222
text{* A consequence: Extensionality. *}
kleing@25610
  1223
wenzelm@26145
  1224
lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
nipkow@26178
  1225
apply (rule iffI)
nipkow@26178
  1226
 prefer 2
nipkow@26178
  1227
 apply clarsimp 
nipkow@26178
  1228
apply (induct A arbitrary: B rule: full_multiset_induct)
nipkow@26178
  1229
apply (rename_tac C)
nipkow@26178
  1230
apply (case_tac B rule: multiset_cases)
nipkow@26178
  1231
 apply (simp add: empty_multiset_count)
nipkow@26178
  1232
apply simp
nipkow@26178
  1233
apply (case_tac "x \<in># C")
nipkow@26178
  1234
 apply (force dest: multi_member_split)
nipkow@26178
  1235
apply (erule_tac x = x in allE)
nipkow@26178
  1236
apply simp
nipkow@26178
  1237
done
kleing@25610
  1238
kleing@25610
  1239
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
kleing@25610
  1240
wenzelm@26145
  1241
kleing@25610
  1242
subsection {* The fold combinator *}
kleing@25610
  1243
wenzelm@26145
  1244
text {*
wenzelm@26145
  1245
  The intended behaviour is
wenzelm@26145
  1246
  @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
wenzelm@26145
  1247
  if @{text f} is associative-commutative. 
kleing@25610
  1248
*}
kleing@25610
  1249
wenzelm@26145
  1250
text {*
wenzelm@26145
  1251
  The graph of @{text "fold_mset"}, @{text "z"}: the start element,
wenzelm@26145
  1252
  @{text "f"}: folding function, @{text "A"}: the multiset, @{text
wenzelm@26145
  1253
  "y"}: the result.
wenzelm@26145
  1254
*}
kleing@25610
  1255
inductive 
kleing@25759
  1256
  fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 
kleing@25610
  1257
  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
kleing@25610
  1258
  and z :: 'b
kleing@25610
  1259
where
kleing@25759
  1260
  emptyI [intro]:  "fold_msetG f z {#} z"
kleing@25759
  1261
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
kleing@25610
  1262
kleing@25759
  1263
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
kleing@25759
  1264
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 
kleing@25610
  1265
kleing@25610
  1266
definition
wenzelm@26145
  1267
  fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
wenzelm@26145
  1268
  "fold_mset f z A = (THE x. fold_msetG f z A x)"
kleing@25610
  1269
kleing@25759
  1270
lemma Diff1_fold_msetG:
wenzelm@26145
  1271
  "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
nipkow@26178
  1272
apply (frule_tac x = x in fold_msetG.insertI)
nipkow@26178
  1273
apply auto
nipkow@26178
  1274
done
kleing@25610
  1275
kleing@25759
  1276
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
nipkow@26178
  1277
apply (induct A)
nipkow@26178
  1278
 apply blast
nipkow@26178
  1279
apply clarsimp
nipkow@26178
  1280
apply (drule_tac x = x in fold_msetG.insertI)
nipkow@26178
  1281
apply auto
nipkow@26178
  1282
done
kleing@25610
  1283
kleing@25759
  1284
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
nipkow@26178
  1285
unfolding fold_mset_def by blast
kleing@25610
  1286
kleing@25610
  1287
locale left_commutative = 
nipkow@26178
  1288
fixes f :: "'a => 'b => 'b"
nipkow@26178
  1289
assumes left_commute: "f x (f y z) = f y (f x z)"
wenzelm@26145
  1290
begin
kleing@25610
  1291
wenzelm@26145
  1292
lemma fold_msetG_determ:
wenzelm@26145
  1293
  "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
kleing@25610
  1294
proof (induct arbitrary: x y z rule: full_multiset_induct)
kleing@25610
  1295
  case (less M x\<^isub>1 x\<^isub>2 Z)
kleing@25610
  1296
  have IH: "\<forall>A. A \<subset># M \<longrightarrow> 
kleing@25759
  1297
    (\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
kleing@25610
  1298
               \<longrightarrow> x' = x)" by fact
kleing@25759
  1299
  have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+
kleing@25610
  1300
  show ?case
kleing@25759
  1301
  proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
kleing@25610
  1302
    assume "M = {#}" and "x\<^isub>1 = Z"
wenzelm@26145
  1303
    then show ?case using Mfoldx\<^isub>2 by auto 
kleing@25610
  1304
  next
kleing@25610
  1305
    fix B b u
kleing@25759
  1306
    assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
wenzelm@26145
  1307
    then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
kleing@25610
  1308
    show ?case
kleing@25759
  1309
    proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
kleing@25610
  1310
      assume "M = {#}" "x\<^isub>2 = Z"
wenzelm@26145
  1311
      then show ?case using Mfoldx\<^isub>1 by auto
kleing@25610
  1312
    next
kleing@25610
  1313
      fix C c v
kleing@25759
  1314
      assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
wenzelm@26145
  1315
      then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
wenzelm@26145
  1316
      then have CsubM: "C \<subset># M" by simp
kleing@25610
  1317
      from MBb have BsubM: "B \<subset># M" by simp
kleing@25610
  1318
      show ?case
kleing@25610
  1319
      proof cases
kleing@25610
  1320
        assume "b=c"
kleing@25610
  1321
        then moreover have "B = C" using MBb MCc by auto
kleing@25610
  1322
        ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
kleing@25610
  1323
      next
kleing@25610
  1324
        assume diff: "b \<noteq> c"
kleing@25610
  1325
        let ?D = "B - {#c#}"
kleing@25610
  1326
        have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
kleing@25610
  1327
          by (auto intro: insert_noteq_member dest: sym)
kleing@25610
  1328
        have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
wenzelm@26145
  1329
        then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
kleing@25610
  1330
        from MBb MCc have "B + {#b#} = C + {#c#}" by blast
wenzelm@26145
  1331
        then have [simp]: "B + {#b#} - {#c#} = C"
kleing@25610
  1332
          using MBb MCc binC cinB by auto
kleing@25610
  1333
        have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
kleing@25610
  1334
          using MBb MCc diff binC cinB
kleing@25610
  1335
          by (auto simp: multiset_add_sub_el_shuffle)
kleing@25759
  1336
        then obtain d where Dfoldd: "fold_msetG f Z ?D d"
kleing@25759
  1337
          using fold_msetG_nonempty by iprover
wenzelm@26145
  1338
        then have "fold_msetG f Z B (f c d)" using cinB
kleing@25759
  1339
          by (rule Diff1_fold_msetG)
wenzelm@26145
  1340
        then have "f c d = u" using IH BsubM Bu by blast
kleing@25610
  1341
        moreover 
kleing@25759
  1342
        have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
kleing@25610
  1343
          by (auto simp: multiset_add_sub_el_shuffle 
kleing@25759
  1344
            dest: fold_msetG.insertI [where x=b])
wenzelm@26145
  1345
        then have "f b d = v" using IH CsubM Cv by blast
kleing@25610
  1346
        ultimately show ?thesis using x\<^isub>1 x\<^isub>2
kleing@25610
  1347
          by (auto simp: left_commute)
kleing@25610
  1348
      qed
kleing@25610
  1349
    qed
kleing@25610
  1350
  qed
kleing@25610
  1351
qed
kleing@25610
  1352
        
wenzelm@26145
  1353
lemma fold_mset_insert_aux:
wenzelm@26145
  1354
  "(fold_msetG f z (A + {#x#}) v) =
kleing@25759
  1355
    (\<exists>y. fold_msetG f z A y \<and> v = f x y)"
nipkow@26178
  1356
apply (rule iffI)
nipkow@26178
  1357
 prefer 2
nipkow@26178
  1358
 apply blast
nipkow@26178
  1359
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
nipkow@26178
  1360
apply (blast intro: fold_msetG_determ)
nipkow@26178
  1361
done
kleing@25610
  1362
wenzelm@26145
  1363
lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
nipkow@26178
  1364
unfolding fold_mset_def by (blast intro: fold_msetG_determ)
kleing@25610
  1365
wenzelm@26145
  1366
lemma fold_mset_insert:
nipkow@26178
  1367
  "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
nipkow@26178
  1368
apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)  
nipkow@26178
  1369
apply (rule the_equality)
nipkow@26178
  1370
 apply (auto cong add: conj_cong 
wenzelm@26145
  1371
     simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
nipkow@26178
  1372
done
kleing@25759
  1373
wenzelm@26145
  1374
lemma fold_mset_insert_idem:
nipkow@26178
  1375
  "fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"
nipkow@26178
  1376
apply (simp add: fold_mset_def fold_mset_insert_aux)
nipkow@26178
  1377
apply (rule the_equality)
nipkow@26178
  1378
 apply (auto cong add: conj_cong 
wenzelm@26145
  1379
     simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
nipkow@26178
  1380
done
kleing@25610
  1381
wenzelm@26145
  1382
lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
nipkow@26178
  1383
by (induct A) (auto simp: fold_mset_insert left_commute [of x])
nipkow@26178
  1384
wenzelm@26145
  1385
lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
nipkow@26178
  1386
using fold_mset_insert [of z "{#}"] by simp
kleing@25610
  1387
wenzelm@26145
  1388
lemma fold_mset_union [simp]:
wenzelm@26145
  1389
  "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
kleing@25759
  1390
proof (induct A)
wenzelm@26145
  1391
  case empty then show ?case by simp
kleing@25759
  1392
next
wenzelm@26145
  1393
  case (add A x)
wenzelm@26145
  1394
  have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
wenzelm@26145
  1395
  then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" 
wenzelm@26145
  1396
    by (simp add: fold_mset_insert)
wenzelm@26145
  1397
  also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
wenzelm@26145
  1398
    by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
wenzelm@26145
  1399
  finally show ?case .
kleing@25759
  1400
qed
kleing@25759
  1401
wenzelm@26145
  1402
lemma fold_mset_fusion:
ballarin@27611
  1403
  assumes "left_commutative g"
ballarin@27611
  1404
  shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
ballarin@27611
  1405
proof -
ballarin@29233
  1406
  interpret left_commutative g by fact
ballarin@27611
  1407
  show "PROP ?P" by (induct A) auto
ballarin@27611
  1408
qed
kleing@25610
  1409
wenzelm@26145
  1410
lemma fold_mset_rec:
wenzelm@26145
  1411
  assumes "a \<in># A" 
kleing@25759
  1412
  shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
kleing@25610
  1413
proof -
wenzelm@26145
  1414
  from assms obtain A' where "A = A' + {#a#}"
wenzelm@26145
  1415
    by (blast dest: multi_member_split)
wenzelm@26145
  1416
  then show ?thesis by simp
kleing@25610
  1417
qed
kleing@25610
  1418
wenzelm@26145
  1419
end
wenzelm@26145
  1420
wenzelm@26145
  1421
text {*
wenzelm@26145
  1422
  A note on code generation: When defining some function containing a
wenzelm@26145
  1423
  subterm @{term"fold_mset F"}, code generation is not automatic. When
wenzelm@26145
  1424
  interpreting locale @{text left_commutative} with @{text F}, the
wenzelm@26145
  1425
  would be code thms for @{const fold_mset} become thms like
wenzelm@26145
  1426
  @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
wenzelm@26145
  1427
  contains defined symbols, i.e.\ is not a code thm. Hence a separate
wenzelm@26145
  1428
  constant with its own code thms needs to be introduced for @{text
wenzelm@26145
  1429
  F}. See the image operator below.
wenzelm@26145
  1430
*}
wenzelm@26145
  1431
nipkow@26016
  1432
nipkow@26016
  1433
subsection {* Image *}
nipkow@26016
  1434
haftmann@28708
  1435
definition [code del]:
haftmann@28708
  1436
 "image_mset f = fold_mset (op + o single o f) {#}"
nipkow@26016
  1437
ballarin@29233
  1438
interpretation image_left_comm!: left_commutative "op + o single o f"
haftmann@28823
  1439
  proof qed (simp add:union_ac)
nipkow@26016
  1440
haftmann@28708
  1441
lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
nipkow@26178
  1442
by (simp add: image_mset_def)
nipkow@26016
  1443
haftmann@28708
  1444
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
nipkow@26178
  1445
by (simp add: image_mset_def)
nipkow@26016
  1446
nipkow@26016
  1447
lemma image_mset_insert:
nipkow@26016
  1448
  "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
nipkow@26178
  1449
by (simp add: image_mset_def add_ac)
nipkow@26016
  1450
haftmann@28708
  1451
lemma image_mset_union [simp]:
nipkow@26016
  1452
  "image_mset f (M+N) = image_mset f M + image_mset f N"
nipkow@26178
  1453
apply (induct N)
nipkow@26178
  1454
 apply simp
nipkow@26178
  1455
apply (simp add: union_assoc [symmetric] image_mset_insert)
nipkow@26178
  1456
done
nipkow@26016
  1457
wenzelm@26145
  1458
lemma size_image_mset [simp]: "size (image_mset f M) = size M"
nipkow@26178
  1459
by (induct M) simp_all
nipkow@26016
  1460
wenzelm@26145
  1461
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
nipkow@26178
  1462
by (cases M) auto
nipkow@26016
  1463
wenzelm@26145
  1464
syntax
wenzelm@26145
  1465
  comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
wenzelm@26145
  1466
      ("({#_/. _ :# _#})")
wenzelm@26145
  1467
translations
wenzelm@26145
  1468
  "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
nipkow@26016
  1469
wenzelm@26145
  1470
syntax
wenzelm@26145
  1471
  comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
wenzelm@26145
  1472
      ("({#_/ | _ :# _./ _#})")
nipkow@26016
  1473
translations
nipkow@26033
  1474
  "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
nipkow@26016
  1475
wenzelm@26145
  1476
text {*
wenzelm@26145
  1477
  This allows to write not just filters like @{term "{#x:#M. x<c#}"}
wenzelm@26145
  1478
  but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
wenzelm@26145
  1479
  "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
wenzelm@26145
  1480
  @{term "{#x+x|x:#M. x<c#}"}.
wenzelm@26145
  1481
*}
nipkow@26016
  1482
krauss@29125
  1483
krauss@29125
  1484
subsection {* Termination proofs with multiset orders *}
krauss@29125
  1485
krauss@29125
  1486
lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
krauss@29125
  1487
  and multi_member_this: "x \<in># {# x #} + XS"
krauss@29125
  1488
  and multi_member_last: "x \<in># {# x #}"
krauss@29125
  1489
  by auto
krauss@29125
  1490
krauss@29125
  1491
definition "ms_strict = mult pair_less"
krauss@29125
  1492
definition "ms_weak = ms_strict \<union> Id"
krauss@29125
  1493
krauss@29125
  1494
lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
krauss@29125
  1495
unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
krauss@29125
  1496
by (auto intro: wf_mult1 wf_trancl simp: mult_def)
krauss@29125
  1497
krauss@29125
  1498
lemma smsI:
krauss@29125
  1499
  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
krauss@29125
  1500
  unfolding ms_strict_def
krauss@29125
  1501
by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
krauss@29125
  1502
krauss@29125
  1503
lemma wmsI:
krauss@29125
  1504
  "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
krauss@29125
  1505
  \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
krauss@29125
  1506
unfolding ms_weak_def ms_strict_def
krauss@29125
  1507
by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
krauss@29125
  1508
krauss@29125
  1509
inductive pw_leq
krauss@29125
  1510
where
krauss@29125
  1511
  pw_leq_empty: "pw_leq {#} {#}"
krauss@29125
  1512
| pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
krauss@29125
  1513
krauss@29125
  1514
lemma pw_leq_lstep:
krauss@29125
  1515
  "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
krauss@29125
  1516
by (drule pw_leq_step) (rule pw_leq_empty, simp)
krauss@29125
  1517
krauss@29125
  1518
lemma pw_leq_split:
krauss@29125
  1519
  assumes "pw_leq X Y"
krauss@29125
  1520
  shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125
  1521
  using assms
krauss@29125
  1522
proof (induct)
krauss@29125
  1523
  case pw_leq_empty thus ?case by auto
krauss@29125
  1524
next
krauss@29125
  1525
  case (pw_leq_step x y X Y)
krauss@29125
  1526
  then obtain A B Z where
krauss@29125
  1527
    [simp]: "X = A + Z" "Y = B + Z" 
krauss@29125
  1528
      and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})" 
krauss@29125
  1529
    by auto
krauss@29125
  1530
  from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less" 
krauss@29125
  1531
    unfolding pair_leq_def by auto
krauss@29125
  1532
  thus ?case
krauss@29125
  1533
  proof
krauss@29125
  1534
    assume [simp]: "x = y"
krauss@29125
  1535
    have
krauss@29125
  1536
      "{#x#} + X = A + ({#y#}+Z) 
krauss@29125
  1537
      \<and> {#y#} + Y = B + ({#y#}+Z)
krauss@29125
  1538
      \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125
  1539
      by (auto simp: add_ac)
krauss@29125
  1540
    thus ?case by (intro exI)
krauss@29125
  1541
  next
krauss@29125
  1542
    assume A: "(x, y) \<in> pair_less"
krauss@29125
  1543
    let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
krauss@29125
  1544
    have "{#x#} + X = ?A' + Z"
krauss@29125
  1545
      "{#y#} + Y = ?B' + Z"
krauss@29125
  1546
      by (auto simp add: add_ac)
krauss@29125
  1547
    moreover have 
krauss@29125
  1548
      "(set_of ?A', set_of ?B') \<in> max_strict"
krauss@29125
  1549
      using 1 A unfolding max_strict_def 
krauss@29125
  1550
      by (auto elim!: max_ext.cases)
krauss@29125
  1551
    ultimately show ?thesis by blast
krauss@29125
  1552
  qed
krauss@29125
  1553
qed
krauss@29125
  1554
krauss@29125
  1555
lemma 
krauss@29125
  1556
  assumes pwleq: "pw_leq Z Z'"
krauss@29125
  1557
  shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
krauss@29125
  1558
  and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
krauss@29125
  1559
  and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
krauss@29125
  1560
proof -
krauss@29125
  1561
  from pw_leq_split[OF pwleq] 
krauss@29125
  1562
  obtain A' B' Z''
krauss@29125
  1563
    where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
krauss@29125
  1564
    and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
krauss@29125
  1565
    by blast
krauss@29125
  1566
  {
krauss@29125
  1567
    assume max: "(set_of A, set_of B) \<in> max_strict"
krauss@29125
  1568
    from mx_or_empty
krauss@29125
  1569
    have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
krauss@29125
  1570
    proof
krauss@29125
  1571
      assume max': "(set_of A', set_of B') \<in> max_strict"
krauss@29125
  1572
      with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
krauss@29125
  1573
        by (auto simp: max_strict_def intro: max_ext_additive)
krauss@29125
  1574
      thus ?thesis by (rule smsI) 
krauss@29125
  1575
    next
krauss@29125
  1576
      assume [simp]: "A' = {#} \<and> B' = {#}"
krauss@29125
  1577
      show ?thesis by (rule smsI) (auto intro: max)
krauss@29125
  1578
    qed
krauss@29125
  1579
    thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
krauss@29125
  1580
    thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
krauss@29125
  1581
  }
krauss@29125
  1582
  from mx_or_empty
krauss@29125
  1583
  have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
krauss@29125
  1584
  thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
krauss@29125
  1585
qed
krauss@29125
  1586
krauss@29125
  1587
lemma empty_idemp: "{#} + x = x" "x + {#} = x"
krauss@29125
  1588
and nonempty_plus: "{# x #} + rs \<noteq> {#}"
krauss@29125
  1589
and nonempty_single: "{# x #} \<noteq> {#}"
krauss@29125
  1590
by auto
krauss@29125
  1591
krauss@29125
  1592
setup {*
krauss@29125
  1593
let
krauss@29125
  1594
  fun msetT T = Type ("Multiset.multiset", [T]);
krauss@29125
  1595
krauss@29125
  1596
  fun mk_mset T [] = Const (@{const_name Mempty}, msetT T)
krauss@29125
  1597
    | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
krauss@29125
  1598
    | mk_mset T (x :: xs) =
krauss@29125
  1599
          Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
krauss@29125
  1600
                mk_mset T [x] $ mk_mset T xs
krauss@29125
  1601
krauss@29125
  1602
  fun mset_member_tac m i =
krauss@29125
  1603
      (if m <= 0 then
krauss@29125
  1604
           rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
krauss@29125
  1605
       else
krauss@29125
  1606
           rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
krauss@29125
  1607
krauss@29125
  1608
  val mset_nonempty_tac =
krauss@29125
  1609
      rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
krauss@29125
  1610
krauss@29125
  1611
  val regroup_munion_conv =
krauss@29125
  1612
      FundefLib.regroup_conv @{const_name Multiset.Mempty} @{const_name plus}
krauss@29125
  1613
        (map (fn t => t RS eq_reflection) (@{thms union_ac} @ @{thms empty_idemp}))
krauss@29125
  1614
krauss@29125
  1615
  fun unfold_pwleq_tac i =
krauss@29125
  1616
    (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
krauss@29125
  1617
      ORELSE (rtac @{thm pw_leq_lstep} i)
krauss@29125
  1618
      ORELSE (rtac @{thm pw_leq_empty} i)
krauss@29125
  1619
krauss@29125
  1620
  val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
krauss@29125
  1621
                      @{thm Un_insert_left}, @{thm Un_empty_left}]
krauss@29125
  1622
in
krauss@29125
  1623
  ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset 
krauss@29125
  1624
  {
krauss@29125
  1625
    msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
krauss@29125
  1626
    mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
krauss@29125
  1627
    mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
krauss@29125
  1628
    smsI'=@{thm ms_strictI}, wmsI2''=@{thm ms_weakI2}, wmsI1=@{thm ms_weakI1},
krauss@29125
  1629
    reduction_pair=@{thm ms_reduction_pair}
krauss@29125
  1630
  })
wenzelm@10249
  1631
end
krauss@29125
  1632
*}
krauss@29125
  1633
krauss@29125
  1634
end