src/HOL/Library/Multiset.thy
author haftmann
Wed Oct 29 11:33:40 2008 +0100 (2008-10-29)
changeset 28708 a1a436f09ec6
parent 28562 4e74209f113e
child 28823 dcbef866c9e2
permissions -rw-r--r--
explicit check for pattern discipline before code translation
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(*  Title:      HOL/Library/Multiset.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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*)
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header {* Multisets *}
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theory Multiset
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imports 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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   330
by (metis count_empty multiset_eq_conv_count_eq)
nipkow@26016
   331
nipkow@26016
   332
kleing@15869
   333
subsubsection {* Intersection *}
kleing@15869
   334
kleing@15869
   335
lemma multiset_inter_count:
nipkow@26178
   336
  "count (A #\<inter> B) x = min (count A x) (count B x)"
nipkow@26178
   337
by (simp add: multiset_inter_def min_def)
kleing@15869
   338
kleing@15869
   339
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
nipkow@26178
   340
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214
   341
    min_max.inf_commute)
kleing@15869
   342
kleing@15869
   343
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
nipkow@26178
   344
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214
   345
    min_max.inf_assoc)
kleing@15869
   346
kleing@15869
   347
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
nipkow@26178
   348
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
kleing@15869
   349
wenzelm@17161
   350
lemmas multiset_inter_ac =
wenzelm@17161
   351
  multiset_inter_commute
wenzelm@17161
   352
  multiset_inter_assoc
wenzelm@17161
   353
  multiset_inter_left_commute
kleing@15869
   354
bulwahn@26143
   355
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
nipkow@26178
   356
by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
bulwahn@26143
   357
kleing@15869
   358
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
nipkow@26178
   359
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
wenzelm@17161
   360
    split: split_if_asm)
nipkow@26178
   361
apply clarsimp
nipkow@26178
   362
apply (erule_tac x = a in allE)
nipkow@26178
   363
apply auto
nipkow@26178
   364
done
kleing@15869
   365
wenzelm@10249
   366
nipkow@26016
   367
subsubsection {* Comprehension (filter) *}
nipkow@26016
   368
haftmann@28708
   369
lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
nipkow@26178
   370
by (simp add: MCollect_def Mempty_def Abs_multiset_inject
wenzelm@26145
   371
    in_multiset expand_fun_eq)
nipkow@26016
   372
haftmann@28708
   373
lemma MCollect_single [simp]:
nipkow@26178
   374
  "MCollect {#x#} P = (if P x then {#x#} else {#})"
nipkow@26178
   375
by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
wenzelm@26145
   376
    in_multiset expand_fun_eq)
nipkow@26016
   377
haftmann@28708
   378
lemma MCollect_union [simp]:
nipkow@26016
   379
  "MCollect (M+N) f = MCollect M f + MCollect N f"
nipkow@26178
   380
by (simp add: MCollect_def union_def Abs_multiset_inject
wenzelm@26145
   381
    in_multiset expand_fun_eq)
nipkow@26016
   382
nipkow@26016
   383
nipkow@26016
   384
subsection {* Induction and case splits *}
wenzelm@10249
   385
wenzelm@10249
   386
lemma setsum_decr:
wenzelm@11701
   387
  "finite F ==> (0::nat) < f a ==>
paulson@15072
   388
    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
nipkow@26178
   389
apply (induct rule: finite_induct)
nipkow@26178
   390
 apply auto
nipkow@26178
   391
apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@26178
   392
done
wenzelm@10249
   393
wenzelm@10313
   394
lemma rep_multiset_induct_aux:
nipkow@26178
   395
assumes 1: "P (\<lambda>a. (0::nat))"
nipkow@26178
   396
  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow@26178
   397
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow@26178
   398
apply (unfold multiset_def)
nipkow@26178
   399
apply (induct_tac n, simp, clarify)
nipkow@26178
   400
 apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow@26178
   401
  apply simp
nipkow@26178
   402
  apply (rule 1)
nipkow@26178
   403
 apply (rule ext, force, clarify)
nipkow@26178
   404
apply (frule setsum_SucD, clarify)
nipkow@26178
   405
apply (rename_tac a)
nipkow@26178
   406
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow@26178
   407
 prefer 2
nipkow@26178
   408
 apply (rule finite_subset)
nipkow@26178
   409
  prefer 2
nipkow@26178
   410
  apply assumption
nipkow@26178
   411
 apply simp
nipkow@26178
   412
 apply blast
nipkow@26178
   413
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow@26178
   414
 prefer 2
nipkow@26178
   415
 apply (rule ext)
nipkow@26178
   416
 apply (simp (no_asm_simp))
nipkow@26178
   417
 apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow@26178
   418
apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow@26178
   419
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow@26178
   420
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow@26178
   421
 prefer 2
nipkow@26178
   422
 apply blast
nipkow@26178
   423
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow@26178
   424
 prefer 2
nipkow@26178
   425
 apply blast
nipkow@26178
   426
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow@26178
   427
done
wenzelm@10249
   428
wenzelm@10313
   429
theorem rep_multiset_induct:
nipkow@11464
   430
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701
   431
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
nipkow@26178
   432
using rep_multiset_induct_aux by blast
wenzelm@10249
   433
wenzelm@18258
   434
theorem multiset_induct [case_names empty add, induct type: multiset]:
nipkow@26178
   435
assumes empty: "P {#}"
nipkow@26178
   436
  and add: "!!M x. P M ==> P (M + {#x#})"
nipkow@26178
   437
shows "P M"
wenzelm@10249
   438
proof -
wenzelm@10249
   439
  note defns = union_def single_def Mempty_def
wenzelm@10249
   440
  show ?thesis
wenzelm@10249
   441
    apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313
   442
    apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@18258
   443
     apply (rule empty [unfolded defns])
paulson@15072
   444
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249
   445
     prefer 2
wenzelm@10249
   446
     apply (simp add: expand_fun_eq)
wenzelm@10249
   447
    apply (erule ssubst)
wenzelm@17200
   448
    apply (erule Abs_multiset_inverse [THEN subst])
nipkow@26016
   449
    apply (drule add [unfolded defns, simplified])
nipkow@26016
   450
    apply(simp add:in_multiset)
wenzelm@10249
   451
    done
wenzelm@10249
   452
qed
wenzelm@10249
   453
kleing@25610
   454
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
nipkow@26178
   455
by (induct M) auto
kleing@25610
   456
kleing@25610
   457
lemma multiset_cases [cases type, case_names empty add]:
nipkow@26178
   458
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow@26178
   459
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow@26178
   460
shows "P"
kleing@25610
   461
proof (cases "M = {#}")
wenzelm@26145
   462
  assume "M = {#}" then show ?thesis using em by simp
kleing@25610
   463
next
kleing@25610
   464
  assume "M \<noteq> {#}"
kleing@25610
   465
  then obtain M' m where "M = M' + {#m#}" 
kleing@25610
   466
    by (blast dest: multi_nonempty_split)
wenzelm@26145
   467
  then show ?thesis using add by simp
kleing@25610
   468
qed
kleing@25610
   469
kleing@25610
   470
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
nipkow@26178
   471
apply (cases M)
nipkow@26178
   472
 apply simp
nipkow@26178
   473
apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow@26178
   474
done
kleing@25610
   475
nipkow@26033
   476
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
nipkow@26178
   477
apply (subst multiset_eq_conv_count_eq)
nipkow@26178
   478
apply auto
nipkow@26178
   479
done
wenzelm@10249
   480
kleing@15869
   481
declare multiset_typedef [simp del]
wenzelm@10249
   482
kleing@25610
   483
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
nipkow@26178
   484
by (cases "B = {#}") (auto dest: multi_member_split)
wenzelm@26145
   485
wenzelm@17161
   486
nipkow@26016
   487
subsection {* Orderings *}
wenzelm@10249
   488
wenzelm@10249
   489
subsubsection {* Well-foundedness *}
wenzelm@10249
   490
haftmann@28708
   491
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
haftmann@28708
   492
  [code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
berghofe@23751
   493
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249
   494
haftmann@28708
   495
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
berghofe@23751
   496
  "mult r = (mult1 r)\<^sup>+"
wenzelm@10249
   497
berghofe@23751
   498
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
nipkow@26178
   499
by (simp add: mult1_def)
wenzelm@10249
   500
berghofe@23751
   501
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
berghofe@23751
   502
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
berghofe@23751
   503
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@19582
   504
  (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
wenzelm@10249
   505
proof (unfold mult1_def)
berghofe@23751
   506
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   507
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
berghofe@23751
   508
  let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249
   509
berghofe@23751
   510
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
wenzelm@18258
   511
  then have "\<exists>a' M0' K.
nipkow@11464
   512
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
wenzelm@18258
   513
  then show "?case1 \<or> ?case2"
wenzelm@10249
   514
  proof (elim exE conjE)
wenzelm@10249
   515
    fix a' M0' K
wenzelm@10249
   516
    assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249
   517
    assume "M0 + {#a#} = M0' + {#a'#}"
wenzelm@18258
   518
    then have "M0 = M0' \<and> a = a' \<or>
nipkow@11464
   519
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249
   520
      by (simp only: add_eq_conv_ex)
wenzelm@18258
   521
    then show ?thesis
wenzelm@10249
   522
    proof (elim disjE conjE exE)
wenzelm@10249
   523
      assume "M0 = M0'" "a = a'"
nipkow@11464
   524
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@18258
   525
      then have ?case2 .. then show ?thesis ..
wenzelm@10249
   526
    next
wenzelm@10249
   527
      fix K'
wenzelm@10249
   528
      assume "M0' = K' + {#a#}"
wenzelm@10249
   529
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249
   530
wenzelm@10249
   531
      assume "M0 = K' + {#a'#}"
wenzelm@10249
   532
      with r have "?R (K' + K) M0" by blast
wenzelm@18258
   533
      with n have ?case1 by simp then show ?thesis ..
wenzelm@10249
   534
    qed
wenzelm@10249
   535
  qed
wenzelm@10249
   536
qed
wenzelm@10249
   537
berghofe@23751
   538
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249
   539
proof
wenzelm@10249
   540
  let ?R = "mult1 r"
wenzelm@10249
   541
  let ?W = "acc ?R"
wenzelm@10249
   542
  {
wenzelm@10249
   543
    fix M M0 a
berghofe@23751
   544
    assume M0: "M0 \<in> ?W"
berghofe@23751
   545
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751
   546
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
berghofe@23751
   547
    have "M0 + {#a#} \<in> ?W"
berghofe@23751
   548
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
   549
      fix N
berghofe@23751
   550
      assume "(N, M0 + {#a#}) \<in> ?R"
berghofe@23751
   551
      then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
berghofe@23751
   552
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249
   553
        by (rule less_add)
berghofe@23751
   554
      then show "N \<in> ?W"
wenzelm@10249
   555
      proof (elim exE disjE conjE)
berghofe@23751
   556
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
berghofe@23751
   557
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
berghofe@23751
   558
        from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
berghofe@23751
   559
        then show "N \<in> ?W" by (simp only: N)
wenzelm@10249
   560
      next
wenzelm@10249
   561
        fix K
wenzelm@10249
   562
        assume N: "N = M0 + K"
berghofe@23751
   563
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
berghofe@23751
   564
        then have "M0 + K \<in> ?W"
wenzelm@10249
   565
        proof (induct K)
wenzelm@18730
   566
          case empty
berghofe@23751
   567
          from M0 show "M0 + {#} \<in> ?W" by simp
wenzelm@18730
   568
        next
wenzelm@18730
   569
          case (add K x)
berghofe@23751
   570
          from add.prems have "(x, a) \<in> r" by simp
berghofe@23751
   571
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
berghofe@23751
   572
          moreover from add have "M0 + K \<in> ?W" by simp
berghofe@23751
   573
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
berghofe@23751
   574
          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249
   575
        qed
berghofe@23751
   576
        then show "N \<in> ?W" by (simp only: N)
wenzelm@10249
   577
      qed
wenzelm@10249
   578
    qed
wenzelm@10249
   579
  } note tedious_reasoning = this
wenzelm@10249
   580
berghofe@23751
   581
  assume wf: "wf r"
wenzelm@10249
   582
  fix M
berghofe@23751
   583
  show "M \<in> ?W"
wenzelm@10249
   584
  proof (induct M)
berghofe@23751
   585
    show "{#} \<in> ?W"
wenzelm@10249
   586
    proof (rule accI)
berghofe@23751
   587
      fix b assume "(b, {#}) \<in> ?R"
berghofe@23751
   588
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
   589
    qed
wenzelm@10249
   590
berghofe@23751
   591
    fix M a assume "M \<in> ?W"
berghofe@23751
   592
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   593
    proof induct
wenzelm@10249
   594
      fix a
berghofe@23751
   595
      assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751
   596
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   597
      proof
berghofe@23751
   598
        fix M assume "M \<in> ?W"
berghofe@23751
   599
        then show "M + {#a#} \<in> ?W"
wenzelm@23373
   600
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
wenzelm@10249
   601
      qed
wenzelm@10249
   602
    qed
berghofe@23751
   603
    from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
wenzelm@10249
   604
  qed
wenzelm@10249
   605
qed
wenzelm@10249
   606
berghofe@23751
   607
theorem wf_mult1: "wf r ==> wf (mult1 r)"
nipkow@26178
   608
by (rule acc_wfI) (rule all_accessible)
wenzelm@10249
   609
berghofe@23751
   610
theorem wf_mult: "wf r ==> wf (mult r)"
nipkow@26178
   611
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249
   612
wenzelm@10249
   613
wenzelm@10249
   614
subsubsection {* Closure-free presentation *}
wenzelm@10249
   615
wenzelm@10249
   616
(*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249
   617
wenzelm@10249
   618
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
nipkow@26178
   619
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   620
wenzelm@10249
   621
text {* One direction. *}
wenzelm@10249
   622
wenzelm@10249
   623
lemma mult_implies_one_step:
berghofe@23751
   624
  "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464
   625
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
berghofe@23751
   626
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
nipkow@26178
   627
apply (unfold mult_def mult1_def set_of_def)
nipkow@26178
   628
apply (erule converse_trancl_induct, clarify)
nipkow@26178
   629
 apply (rule_tac x = M0 in exI, simp, clarify)
nipkow@26178
   630
apply (case_tac "a :# K")
nipkow@26178
   631
 apply (rule_tac x = I in exI)
nipkow@26178
   632
 apply (simp (no_asm))
nipkow@26178
   633
 apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
nipkow@26178
   634
 apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@26178
   635
 apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow@26178
   636
 apply (simp add: diff_union_single_conv)
nipkow@26178
   637
 apply (simp (no_asm_use) add: trans_def)
nipkow@26178
   638
 apply blast
nipkow@26178
   639
apply (subgoal_tac "a :# I")
nipkow@26178
   640
 apply (rule_tac x = "I - {#a#}" in exI)
nipkow@26178
   641
 apply (rule_tac x = "J + {#a#}" in exI)
nipkow@26178
   642
 apply (rule_tac x = "K + Ka" in exI)
nipkow@26178
   643
 apply (rule conjI)
nipkow@26178
   644
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow@26178
   645
 apply (rule conjI)
nipkow@26178
   646
  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow@26178
   647
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow@26178
   648
 apply (simp (no_asm_use) add: trans_def)
nipkow@26178
   649
 apply blast
nipkow@26178
   650
apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow@26178
   651
 apply simp
nipkow@26178
   652
apply (simp (no_asm))
nipkow@26178
   653
done
wenzelm@10249
   654
wenzelm@10249
   655
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
nipkow@26178
   656
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   657
nipkow@11464
   658
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
nipkow@26178
   659
apply (erule size_eq_Suc_imp_elem [THEN exE])
nipkow@26178
   660
apply (drule elem_imp_eq_diff_union, auto)
nipkow@26178
   661
done
wenzelm@10249
   662
wenzelm@10249
   663
lemma one_step_implies_mult_aux:
berghofe@23751
   664
  "trans r ==>
berghofe@23751
   665
    \<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
   666
      --> (I + K, I + J) \<in> mult r"
nipkow@26178
   667
apply (induct_tac n, auto)
nipkow@26178
   668
apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow@26178
   669
apply (rename_tac "J'", simp)
nipkow@26178
   670
apply (erule notE, auto)
nipkow@26178
   671
apply (case_tac "J' = {#}")
nipkow@26178
   672
 apply (simp add: mult_def)
nipkow@26178
   673
 apply (rule r_into_trancl)
nipkow@26178
   674
 apply (simp add: mult1_def set_of_def, blast)
nipkow@26178
   675
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@26178
   676
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@26178
   677
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
nipkow@26178
   678
apply (erule ssubst)
nipkow@26178
   679
apply (simp add: Ball_def, auto)
nipkow@26178
   680
apply (subgoal_tac
nipkow@26178
   681
  "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
nipkow@26178
   682
    (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow@26178
   683
 prefer 2
nipkow@26178
   684
 apply force
nipkow@26178
   685
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
nipkow@26178
   686
apply (erule trancl_trans)
nipkow@26178
   687
apply (rule r_into_trancl)
nipkow@26178
   688
apply (simp add: mult1_def set_of_def)
nipkow@26178
   689
apply (rule_tac x = a in exI)
nipkow@26178
   690
apply (rule_tac x = "I + J'" in exI)
nipkow@26178
   691
apply (simp add: union_ac)
nipkow@26178
   692
done
wenzelm@10249
   693
wenzelm@17161
   694
lemma one_step_implies_mult:
berghofe@23751
   695
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
berghofe@23751
   696
    ==> (I + K, I + J) \<in> mult r"
nipkow@26178
   697
using one_step_implies_mult_aux by blast
wenzelm@10249
   698
wenzelm@10249
   699
wenzelm@10249
   700
subsubsection {* Partial-order properties *}
wenzelm@10249
   701
haftmann@26567
   702
instantiation multiset :: (order) order
haftmann@26567
   703
begin
wenzelm@10249
   704
haftmann@28708
   705
definition less_multiset_def [code del]:
haftmann@28708
   706
  "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
haftmann@26567
   707
haftmann@28708
   708
definition le_multiset_def [code del]:
haftmann@28708
   709
  "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249
   710
berghofe@23751
   711
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
nipkow@26178
   712
unfolding trans_def by (blast intro: order_less_trans)
wenzelm@10249
   713
wenzelm@10249
   714
text {*
wenzelm@10249
   715
 \medskip Irreflexivity.
wenzelm@10249
   716
*}
wenzelm@10249
   717
wenzelm@10249
   718
lemma mult_irrefl_aux:
nipkow@26178
   719
  "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
nipkow@26178
   720
by (induct rule: finite_induct) (auto intro: order_less_trans)
wenzelm@10249
   721
wenzelm@17161
   722
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
nipkow@26178
   723
apply (unfold less_multiset_def, auto)
nipkow@26178
   724
apply (drule trans_base_order [THEN mult_implies_one_step], auto)
nipkow@26178
   725
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
nipkow@26178
   726
apply (simp add: set_of_eq_empty_iff)
nipkow@26178
   727
done
wenzelm@10249
   728
wenzelm@10249
   729
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
nipkow@26178
   730
using insert mult_less_not_refl by fast
wenzelm@10249
   731
wenzelm@10249
   732
wenzelm@10249
   733
text {* Transitivity. *}
wenzelm@10249
   734
wenzelm@10249
   735
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
nipkow@26178
   736
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@10249
   737
wenzelm@10249
   738
text {* Asymmetry. *}
wenzelm@10249
   739
nipkow@11464
   740
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
nipkow@26178
   741
apply auto
nipkow@26178
   742
apply (rule mult_less_not_refl [THEN notE])
nipkow@26178
   743
apply (erule mult_less_trans, assumption)
nipkow@26178
   744
done
wenzelm@10249
   745
wenzelm@10249
   746
theorem mult_less_asym:
nipkow@26178
   747
  "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
nipkow@26178
   748
using mult_less_not_sym by blast
wenzelm@10249
   749
wenzelm@10249
   750
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
nipkow@26178
   751
unfolding le_multiset_def by auto
wenzelm@10249
   752
wenzelm@10249
   753
text {* Anti-symmetry. *}
wenzelm@10249
   754
wenzelm@10249
   755
theorem mult_le_antisym:
nipkow@26178
   756
  "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
nipkow@26178
   757
unfolding le_multiset_def by (blast dest: mult_less_not_sym)
wenzelm@10249
   758
wenzelm@10249
   759
text {* Transitivity. *}
wenzelm@10249
   760
wenzelm@10249
   761
theorem mult_le_trans:
nipkow@26178
   762
  "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
nipkow@26178
   763
unfolding le_multiset_def by (blast intro: mult_less_trans)
wenzelm@10249
   764
wenzelm@11655
   765
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
nipkow@26178
   766
unfolding le_multiset_def by auto
wenzelm@10249
   767
haftmann@27682
   768
instance proof
haftmann@27682
   769
qed (auto simp add: mult_less_le dest: mult_le_antisym elim: mult_le_trans)
wenzelm@10277
   770
haftmann@26567
   771
end
haftmann@26567
   772
wenzelm@10249
   773
wenzelm@10249
   774
subsubsection {* Monotonicity of multiset union *}
wenzelm@10249
   775
wenzelm@17161
   776
lemma mult1_union:
nipkow@26178
   777
  "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
nipkow@26178
   778
apply (unfold mult1_def)
nipkow@26178
   779
apply auto
nipkow@26178
   780
apply (rule_tac x = a in exI)
nipkow@26178
   781
apply (rule_tac x = "C + M0" in exI)
nipkow@26178
   782
apply (simp add: union_assoc)
nipkow@26178
   783
done
wenzelm@10249
   784
wenzelm@10249
   785
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
nipkow@26178
   786
apply (unfold less_multiset_def mult_def)
nipkow@26178
   787
apply (erule trancl_induct)
nipkow@26178
   788
 apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
nipkow@26178
   789
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
nipkow@26178
   790
done
wenzelm@10249
   791
wenzelm@10249
   792
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
nipkow@26178
   793
apply (subst union_commute [of B C])
nipkow@26178
   794
apply (subst union_commute [of D C])
nipkow@26178
   795
apply (erule union_less_mono2)
nipkow@26178
   796
done
wenzelm@10249
   797
wenzelm@17161
   798
lemma union_less_mono:
nipkow@26178
   799
  "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
nipkow@26178
   800
by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249
   801
wenzelm@17161
   802
lemma union_le_mono:
nipkow@26178
   803
  "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
nipkow@26178
   804
unfolding le_multiset_def
nipkow@26178
   805
by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249
   806
wenzelm@17161
   807
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
nipkow@26178
   808
apply (unfold le_multiset_def less_multiset_def)
nipkow@26178
   809
apply (case_tac "M = {#}")
nipkow@26178
   810
 prefer 2
nipkow@26178
   811
 apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
nipkow@26178
   812
  prefer 2
nipkow@26178
   813
  apply (rule one_step_implies_mult)
nipkow@26178
   814
    apply (simp only: trans_def)
nipkow@26178
   815
    apply auto
nipkow@26178
   816
done
wenzelm@10249
   817
wenzelm@17161
   818
lemma union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072
   819
proof -
wenzelm@17200
   820
  have "A + {#} <= A + B" by (blast intro: union_le_mono)
wenzelm@18258
   821
  then show ?thesis by simp
paulson@15072
   822
qed
paulson@15072
   823
wenzelm@17161
   824
lemma union_upper2: "B <= A + (B::'a::order multiset)"
nipkow@26178
   825
by (subst union_commute) (rule union_upper1)
paulson@15072
   826
nipkow@23611
   827
instance multiset :: (order) pordered_ab_semigroup_add
nipkow@26178
   828
apply intro_classes
nipkow@26178
   829
apply (erule union_le_mono[OF mult_le_refl])
nipkow@26178
   830
done
wenzelm@26145
   831
paulson@15072
   832
wenzelm@17200
   833
subsection {* Link with lists *}
paulson@15072
   834
nipkow@26016
   835
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
wenzelm@26145
   836
  "multiset_of [] = {#}" |
wenzelm@26145
   837
  "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072
   838
paulson@15072
   839
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
nipkow@26178
   840
by (induct x) auto
paulson@15072
   841
paulson@15072
   842
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
nipkow@26178
   843
by (induct x) auto
paulson@15072
   844
paulson@15072
   845
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
nipkow@26178
   846
by (induct x) auto
kleing@15867
   847
kleing@15867
   848
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
nipkow@26178
   849
by (induct xs) auto
paulson@15072
   850
wenzelm@18258
   851
lemma multiset_of_append [simp]:
nipkow@26178
   852
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
nipkow@26178
   853
by (induct xs arbitrary: ys) (auto simp: union_ac)
wenzelm@18730
   854
paulson@15072
   855
lemma surj_multiset_of: "surj multiset_of"
nipkow@26178
   856
apply (unfold surj_def)
nipkow@26178
   857
apply (rule allI)
nipkow@26178
   858
apply (rule_tac M = y in multiset_induct)
nipkow@26178
   859
 apply auto
nipkow@26178
   860
apply (rule_tac x = "x # xa" in exI)
nipkow@26178
   861
apply auto
nipkow@26178
   862
done
wenzelm@10249
   863
nipkow@25162
   864
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
nipkow@26178
   865
by (induct x) auto
paulson@15072
   866
wenzelm@17200
   867
lemma distinct_count_atmost_1:
nipkow@26178
   868
  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
nipkow@26178
   869
apply (induct x, simp, rule iffI, simp_all)
nipkow@26178
   870
apply (rule conjI)
nipkow@26178
   871
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
nipkow@26178
   872
apply (erule_tac x = a in allE, simp, clarify)
nipkow@26178
   873
apply (erule_tac x = aa in allE, simp)
nipkow@26178
   874
done
paulson@15072
   875
wenzelm@17200
   876
lemma multiset_of_eq_setD:
kleing@15867
   877
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
nipkow@26178
   878
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
kleing@15867
   879
wenzelm@17200
   880
lemma set_eq_iff_multiset_of_eq_distinct:
wenzelm@26145
   881
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
wenzelm@26145
   882
    (set x = set y) = (multiset_of x = multiset_of y)"
nipkow@26178
   883
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
paulson@15072
   884
wenzelm@17200
   885
lemma set_eq_iff_multiset_of_remdups_eq:
paulson@15072
   886
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
nipkow@26178
   887
apply (rule iffI)
nipkow@26178
   888
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
nipkow@26178
   889
apply (drule distinct_remdups [THEN distinct_remdups
wenzelm@26145
   890
      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
nipkow@26178
   891
apply simp
nipkow@26178
   892
done
wenzelm@10249
   893
wenzelm@18258
   894
lemma multiset_of_compl_union [simp]:
nipkow@26178
   895
  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
nipkow@26178
   896
by (induct xs) (auto simp: union_ac)
paulson@15072
   897
wenzelm@17200
   898
lemma count_filter:
nipkow@26178
   899
  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
nipkow@26178
   900
by (induct xs) auto
kleing@15867
   901
bulwahn@26143
   902
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
nipkow@26178
   903
apply (induct ls arbitrary: i)
nipkow@26178
   904
 apply simp
nipkow@26178
   905
apply (case_tac i)
nipkow@26178
   906
 apply auto
nipkow@26178
   907
done
bulwahn@26143
   908
bulwahn@26143
   909
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
nipkow@26178
   910
by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
bulwahn@26143
   911
bulwahn@26143
   912
lemma multiset_of_eq_length:
nipkow@26178
   913
assumes "multiset_of xs = multiset_of ys"
nipkow@26178
   914
shows "length xs = length ys"
nipkow@26178
   915
using assms
bulwahn@26143
   916
proof (induct arbitrary: ys rule: length_induct)
bulwahn@26143
   917
  case (1 xs ys)
bulwahn@26143
   918
  show ?case
bulwahn@26143
   919
  proof (cases xs)
wenzelm@26145
   920
    case Nil with "1.prems" show ?thesis by simp
bulwahn@26143
   921
  next
bulwahn@26143
   922
    case (Cons x xs')
bulwahn@26143
   923
    note xCons = Cons
bulwahn@26143
   924
    show ?thesis
bulwahn@26143
   925
    proof (cases ys)
bulwahn@26143
   926
      case Nil
wenzelm@26145
   927
      with "1.prems" Cons show ?thesis by simp
bulwahn@26143
   928
    next
bulwahn@26143
   929
      case (Cons y ys')
bulwahn@26143
   930
      have x_in_ys: "x = y \<or> x \<in> set ys'"
bulwahn@26143
   931
      proof (cases "x = y")
wenzelm@26145
   932
	case True then show ?thesis ..
bulwahn@26143
   933
      next
bulwahn@26143
   934
	case False
wenzelm@26145
   935
	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
bulwahn@26143
   936
	with False show ?thesis by (simp add: mem_set_multiset_eq)
bulwahn@26143
   937
      qed
wenzelm@26145
   938
      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
wenzelm@26145
   939
	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
wenzelm@26145
   940
      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
bulwahn@26143
   941
	apply -
bulwahn@26143
   942
	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
bulwahn@26143
   943
	apply fastsimp
bulwahn@26143
   944
	done
wenzelm@26145
   945
      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
wenzelm@26145
   946
      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
bulwahn@26143
   947
      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
bulwahn@26143
   948
    qed
bulwahn@26143
   949
  qed
bulwahn@26143
   950
qed
bulwahn@26143
   951
wenzelm@26145
   952
text {*
wenzelm@26145
   953
  This lemma shows which properties suffice to show that a function
wenzelm@26145
   954
  @{text "f"} with @{text "f xs = ys"} behaves like sort.
wenzelm@26145
   955
*}
wenzelm@26145
   956
lemma properties_for_sort:
wenzelm@26145
   957
  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
bulwahn@26143
   958
proof (induct xs arbitrary: ys)
wenzelm@26145
   959
  case Nil then show ?case by simp
bulwahn@26143
   960
next
bulwahn@26143
   961
  case (Cons x xs)
wenzelm@26145
   962
  then have "x \<in> set ys"
wenzelm@26145
   963
    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
bulwahn@26143
   964
  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
bulwahn@26143
   965
    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
bulwahn@26143
   966
qed
bulwahn@26143
   967
kleing@15867
   968
paulson@15072
   969
subsection {* Pointwise ordering induced by count *}
paulson@15072
   970
haftmann@28708
   971
definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
haftmann@28708
   972
  [code del]: "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
wenzelm@26145
   973
haftmann@28708
   974
definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
haftmann@28708
   975
  [code del]: "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
kleing@25610
   976
wenzelm@26145
   977
notation mset_le  (infix "\<subseteq>#" 50)
wenzelm@26145
   978
notation mset_less  (infix "\<subset>#" 50)
paulson@15072
   979
nipkow@23611
   980
lemma mset_le_refl[simp]: "A \<le># A"
nipkow@26178
   981
unfolding mset_le_def by auto
paulson@15072
   982
wenzelm@26145
   983
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
nipkow@26178
   984
unfolding mset_le_def by (fast intro: order_trans)
paulson@15072
   985
wenzelm@26145
   986
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
nipkow@26178
   987
apply (unfold mset_le_def)
nipkow@26178
   988
apply (rule multiset_eq_conv_count_eq [THEN iffD2])
nipkow@26178
   989
apply (blast intro: order_antisym)
nipkow@26178
   990
done
paulson@15072
   991
wenzelm@26145
   992
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
nipkow@26178
   993
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
nipkow@26178
   994
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
nipkow@26178
   995
done
paulson@15072
   996
nipkow@23611
   997
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
nipkow@26178
   998
unfolding mset_le_def by auto
paulson@15072
   999
nipkow@23611
  1000
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
nipkow@26178
  1001
unfolding mset_le_def by auto
paulson@15072
  1002
nipkow@23611
  1003
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
nipkow@26178
  1004
apply (unfold mset_le_def)
nipkow@26178
  1005
apply auto
nipkow@26178
  1006
apply (erule_tac x = a in allE)+
nipkow@26178
  1007
apply auto
nipkow@26178
  1008
done
paulson@15072
  1009
nipkow@23611
  1010
lemma mset_le_add_left[simp]: "A \<le># A + B"
nipkow@26178
  1011
unfolding mset_le_def by auto
paulson@15072
  1012
nipkow@23611
  1013
lemma mset_le_add_right[simp]: "B \<le># A + B"
nipkow@26178
  1014
unfolding mset_le_def by auto
paulson@15072
  1015
bulwahn@26143
  1016
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
nipkow@26178
  1017
by (simp add: mset_le_def)
bulwahn@26143
  1018
bulwahn@26143
  1019
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
nipkow@26178
  1020
by (simp add: multiset_eq_conv_count_eq mset_le_def)
bulwahn@26143
  1021
bulwahn@26143
  1022
lemma mset_le_multiset_union_diff_commute:
nipkow@26178
  1023
assumes "B \<le># A"
nipkow@26178
  1024
shows "A - B + C = A + C - B"
bulwahn@26143
  1025
proof -
wenzelm@26145
  1026
  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
wenzelm@26145
  1027
  from this obtain D where "A = B + D" ..
wenzelm@26145
  1028
  then show ?thesis
wenzelm@26145
  1029
    apply simp
wenzelm@26145
  1030
    apply (subst union_commute)
wenzelm@26145
  1031
    apply (subst multiset_diff_union_assoc)
wenzelm@26145
  1032
    apply simp
wenzelm@26145
  1033
    apply (simp add: diff_cancel)
wenzelm@26145
  1034
    apply (subst union_assoc)
wenzelm@26145
  1035
    apply (subst union_commute[of "B" _])
wenzelm@26145
  1036
    apply (subst multiset_diff_union_assoc)
wenzelm@26145
  1037
    apply simp
wenzelm@26145
  1038
    apply (simp add: diff_cancel)
wenzelm@26145
  1039
    done
bulwahn@26143
  1040
qed
bulwahn@26143
  1041
nipkow@23611
  1042
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
nipkow@26178
  1043
apply (induct xs)
nipkow@26178
  1044
 apply auto
nipkow@26178
  1045
apply (rule mset_le_trans)
nipkow@26178
  1046
 apply auto
nipkow@26178
  1047
done
nipkow@23611
  1048
wenzelm@26145
  1049
lemma multiset_of_update:
wenzelm@26145
  1050
  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
bulwahn@26143
  1051
proof (induct ls arbitrary: i)
wenzelm@26145
  1052
  case Nil then show ?case by simp
bulwahn@26143
  1053
next
bulwahn@26143
  1054
  case (Cons x xs)
bulwahn@26143
  1055
  show ?case
wenzelm@26145
  1056
  proof (cases i)
wenzelm@26145
  1057
    case 0 then show ?thesis by simp
wenzelm@26145
  1058
  next
wenzelm@26145
  1059
    case (Suc i')
wenzelm@26145
  1060
    with Cons show ?thesis
wenzelm@26145
  1061
      apply simp
wenzelm@26145
  1062
      apply (subst union_assoc)
wenzelm@26145
  1063
      apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
wenzelm@26145
  1064
      apply (subst union_assoc [symmetric])
wenzelm@26145
  1065
      apply simp
wenzelm@26145
  1066
      apply (rule mset_le_multiset_union_diff_commute)
wenzelm@26145
  1067
      apply (simp add: mset_le_single nth_mem_multiset_of)
wenzelm@26145
  1068
      done
bulwahn@26143
  1069
  qed
bulwahn@26143
  1070
qed
bulwahn@26143
  1071
wenzelm@26145
  1072
lemma multiset_of_swap:
wenzelm@26145
  1073
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
wenzelm@26145
  1074
    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
nipkow@26178
  1075
apply (case_tac "i = j")
nipkow@26178
  1076
 apply simp
nipkow@26178
  1077
apply (simp add: multiset_of_update)
nipkow@26178
  1078
apply (subst elem_imp_eq_diff_union[symmetric])
nipkow@26178
  1079
 apply (simp add: nth_mem_multiset_of)
nipkow@26178
  1080
apply simp
nipkow@26178
  1081
done
bulwahn@26143
  1082
wenzelm@26145
  1083
interpretation mset_order: order ["op \<le>#" "op <#"]
haftmann@27682
  1084
proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
haftmann@27682
  1085
  mset_le_trans simp: mset_less_def)
nipkow@23611
  1086
nipkow@23611
  1087
interpretation mset_order_cancel_semigroup:
wenzelm@26145
  1088
    pordered_cancel_ab_semigroup_add ["op +" "op \<le>#" "op <#"]
haftmann@27682
  1089
proof qed (erule mset_le_mono_add [OF mset_le_refl])
nipkow@23611
  1090
nipkow@23611
  1091
interpretation mset_order_semigroup_cancel:
wenzelm@26145
  1092
    pordered_ab_semigroup_add_imp_le ["op +" "op \<le>#" "op <#"]
haftmann@27682
  1093
proof qed simp
paulson@15072
  1094
kleing@25610
  1095
wenzelm@26145
  1096
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
nipkow@26178
  1097
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178
  1098
apply (erule_tac x=x in allE)
nipkow@26178
  1099
apply auto
nipkow@26178
  1100
done
kleing@25610
  1101
wenzelm@26145
  1102
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
nipkow@26178
  1103
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178
  1104
apply (erule_tac x = x in allE)
nipkow@26178
  1105
apply auto
nipkow@26178
  1106
done
kleing@25610
  1107
  
wenzelm@26145
  1108
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
nipkow@26178
  1109
apply (rule conjI)
nipkow@26178
  1110
 apply (simp add: mset_lessD)
nipkow@26178
  1111
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178
  1112
apply safe
nipkow@26178
  1113
 apply (erule_tac x = a in allE)
nipkow@26178
  1114
 apply (auto split: split_if_asm)
nipkow@26178
  1115
done
kleing@25610
  1116
wenzelm@26145
  1117
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
nipkow@26178
  1118
apply (rule conjI)
nipkow@26178
  1119
 apply (simp add: mset_leD)
nipkow@26178
  1120
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
nipkow@26178
  1121
done
kleing@25610
  1122
kleing@25610
  1123
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" 
nipkow@26178
  1124
by (induct A) (auto simp: mset_le_def mset_less_def)
kleing@25610
  1125
kleing@25610
  1126
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
nipkow@26178
  1127
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1128
kleing@25610
  1129
lemma multi_psub_self[simp]: "A \<subset># A = False"
nipkow@26178
  1130
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1131
kleing@25610
  1132
lemma mset_less_add_bothsides:
kleing@25610
  1133
  "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
nipkow@26178
  1134
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1135
kleing@25610
  1136
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
nipkow@26178
  1137
by (auto simp: mset_le_def mset_less_def)
kleing@25610
  1138
kleing@25610
  1139
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
kleing@25610
  1140
proof (induct A arbitrary: B)
kleing@25610
  1141
  case (empty M)
wenzelm@26145
  1142
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
kleing@25610
  1143
  then obtain M' x where "M = M' + {#x#}" 
kleing@25610
  1144
    by (blast dest: multi_nonempty_split)
wenzelm@26145
  1145
  then show ?case by simp
kleing@25610
  1146
next
kleing@25610
  1147
  case (add S x T)
kleing@25610
  1148
  have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
kleing@25610
  1149
  have SxsubT: "S + {#x#} \<subset># T" by fact
wenzelm@26145
  1150
  then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
kleing@25610
  1151
  then obtain T' where T: "T = T' + {#x#}" 
kleing@25610
  1152
    by (blast dest: multi_member_split)
wenzelm@26145
  1153
  then have "S \<subset># T'" using SxsubT 
kleing@25610
  1154
    by (blast intro: mset_less_add_bothsides)
wenzelm@26145
  1155
  then have "size S < size T'" using IH by simp
wenzelm@26145
  1156
  then show ?case using T by simp
kleing@25610
  1157
qed
kleing@25610
  1158
kleing@25610
  1159
lemmas mset_less_trans = mset_order.less_eq_less.less_trans
kleing@25610
  1160
kleing@25610
  1161
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
nipkow@26178
  1162
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
kleing@25610
  1163
wenzelm@26145
  1164
kleing@25610
  1165
subsection {* Strong induction and subset induction for multisets *}
kleing@25610
  1166
nipkow@26016
  1167
text {* Well-foundedness of proper subset operator: *}
kleing@25610
  1168
wenzelm@26145
  1169
text {* proper multiset subset *}
kleing@25610
  1170
definition
wenzelm@26145
  1171
  mset_less_rel :: "('a multiset * 'a multiset) set" where
wenzelm@26145
  1172
  "mset_less_rel = {(A,B). A \<subset># B}"
kleing@25610
  1173
kleing@25610
  1174
lemma multiset_add_sub_el_shuffle: 
wenzelm@26145
  1175
  assumes "c \<in># B" and "b \<noteq> c" 
kleing@25610
  1176
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
kleing@25610
  1177
proof -
wenzelm@26145
  1178
  from `c \<in># B` obtain A where B: "B = A + {#c#}" 
kleing@25610
  1179
    by (blast dest: multi_member_split)
kleing@25610
  1180
  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
wenzelm@26145
  1181
  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
kleing@25610
  1182
    by (simp add: union_ac)
wenzelm@26145
  1183
  then show ?thesis using B by simp
kleing@25610
  1184
qed
kleing@25610
  1185
kleing@25610
  1186
lemma wf_mset_less_rel: "wf mset_less_rel"
nipkow@26178
  1187
apply (unfold mset_less_rel_def)
nipkow@26178
  1188
apply (rule wf_measure [THEN wf_subset, where f1=size])
nipkow@26178
  1189
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
nipkow@26178
  1190
done
kleing@25610
  1191
nipkow@26016
  1192
text {* The induction rules: *}
kleing@25610
  1193
kleing@25610
  1194
lemma full_multiset_induct [case_names less]:
nipkow@26178
  1195
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
nipkow@26178
  1196
shows "P B"
nipkow@26178
  1197
apply (rule wf_mset_less_rel [THEN wf_induct])
nipkow@26178
  1198
apply (rule ih, auto simp: mset_less_rel_def)
nipkow@26178
  1199
done
kleing@25610
  1200
kleing@25610
  1201
lemma multi_subset_induct [consumes 2, case_names empty add]:
nipkow@26178
  1202
assumes "F \<subseteq># A"
nipkow@26178
  1203
  and empty: "P {#}"
nipkow@26178
  1204
  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
nipkow@26178
  1205
shows "P F"
kleing@25610
  1206
proof -
kleing@25610
  1207
  from `F \<subseteq># A`
kleing@25610
  1208
  show ?thesis
kleing@25610
  1209
  proof (induct F)
kleing@25610
  1210
    show "P {#}" by fact
kleing@25610
  1211
  next
kleing@25610
  1212
    fix x F
kleing@25610
  1213
    assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
kleing@25610
  1214
    show "P (F + {#x#})"
kleing@25610
  1215
    proof (rule insert)
kleing@25610
  1216
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
wenzelm@26145
  1217
      from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
kleing@25610
  1218
      with P show "P F" .
kleing@25610
  1219
    qed
kleing@25610
  1220
  qed
kleing@25610
  1221
qed 
kleing@25610
  1222
nipkow@26016
  1223
text{* A consequence: Extensionality. *}
kleing@25610
  1224
wenzelm@26145
  1225
lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
nipkow@26178
  1226
apply (rule iffI)
nipkow@26178
  1227
 prefer 2
nipkow@26178
  1228
 apply clarsimp 
nipkow@26178
  1229
apply (induct A arbitrary: B rule: full_multiset_induct)
nipkow@26178
  1230
apply (rename_tac C)
nipkow@26178
  1231
apply (case_tac B rule: multiset_cases)
nipkow@26178
  1232
 apply (simp add: empty_multiset_count)
nipkow@26178
  1233
apply simp
nipkow@26178
  1234
apply (case_tac "x \<in># C")
nipkow@26178
  1235
 apply (force dest: multi_member_split)
nipkow@26178
  1236
apply (erule_tac x = x in allE)
nipkow@26178
  1237
apply simp
nipkow@26178
  1238
done
kleing@25610
  1239
kleing@25610
  1240
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
kleing@25610
  1241
wenzelm@26145
  1242
kleing@25610
  1243
subsection {* The fold combinator *}
kleing@25610
  1244
wenzelm@26145
  1245
text {*
wenzelm@26145
  1246
  The intended behaviour is
wenzelm@26145
  1247
  @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
wenzelm@26145
  1248
  if @{text f} is associative-commutative. 
kleing@25610
  1249
*}
kleing@25610
  1250
wenzelm@26145
  1251
text {*
wenzelm@26145
  1252
  The graph of @{text "fold_mset"}, @{text "z"}: the start element,
wenzelm@26145
  1253
  @{text "f"}: folding function, @{text "A"}: the multiset, @{text
wenzelm@26145
  1254
  "y"}: the result.
wenzelm@26145
  1255
*}
kleing@25610
  1256
inductive 
kleing@25759
  1257
  fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 
kleing@25610
  1258
  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
kleing@25610
  1259
  and z :: 'b
kleing@25610
  1260
where
kleing@25759
  1261
  emptyI [intro]:  "fold_msetG f z {#} z"
kleing@25759
  1262
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
kleing@25610
  1263
kleing@25759
  1264
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
kleing@25759
  1265
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 
kleing@25610
  1266
kleing@25610
  1267
definition
wenzelm@26145
  1268
  fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
wenzelm@26145
  1269
  "fold_mset f z A = (THE x. fold_msetG f z A x)"
kleing@25610
  1270
kleing@25759
  1271
lemma Diff1_fold_msetG:
wenzelm@26145
  1272
  "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
nipkow@26178
  1273
apply (frule_tac x = x in fold_msetG.insertI)
nipkow@26178
  1274
apply auto
nipkow@26178
  1275
done
kleing@25610
  1276
kleing@25759
  1277
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
nipkow@26178
  1278
apply (induct A)
nipkow@26178
  1279
 apply blast
nipkow@26178
  1280
apply clarsimp
nipkow@26178
  1281
apply (drule_tac x = x in fold_msetG.insertI)
nipkow@26178
  1282
apply auto
nipkow@26178
  1283
done
kleing@25610
  1284
kleing@25759
  1285
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
nipkow@26178
  1286
unfolding fold_mset_def by blast
kleing@25610
  1287
kleing@25610
  1288
locale left_commutative = 
nipkow@26178
  1289
fixes f :: "'a => 'b => 'b"
nipkow@26178
  1290
assumes left_commute: "f x (f y z) = f y (f x z)"
wenzelm@26145
  1291
begin
kleing@25610
  1292
wenzelm@26145
  1293
lemma fold_msetG_determ:
wenzelm@26145
  1294
  "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
kleing@25610
  1295
proof (induct arbitrary: x y z rule: full_multiset_induct)
kleing@25610
  1296
  case (less M x\<^isub>1 x\<^isub>2 Z)
kleing@25610
  1297
  have IH: "\<forall>A. A \<subset># M \<longrightarrow> 
kleing@25759
  1298
    (\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
kleing@25610
  1299
               \<longrightarrow> x' = x)" by fact
kleing@25759
  1300
  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
  1301
  show ?case
kleing@25759
  1302
  proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
kleing@25610
  1303
    assume "M = {#}" and "x\<^isub>1 = Z"
wenzelm@26145
  1304
    then show ?case using Mfoldx\<^isub>2 by auto 
kleing@25610
  1305
  next
kleing@25610
  1306
    fix B b u
kleing@25759
  1307
    assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
wenzelm@26145
  1308
    then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
kleing@25610
  1309
    show ?case
kleing@25759
  1310
    proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
kleing@25610
  1311
      assume "M = {#}" "x\<^isub>2 = Z"
wenzelm@26145
  1312
      then show ?case using Mfoldx\<^isub>1 by auto
kleing@25610
  1313
    next
kleing@25610
  1314
      fix C c v
kleing@25759
  1315
      assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
wenzelm@26145
  1316
      then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
wenzelm@26145
  1317
      then have CsubM: "C \<subset># M" by simp
kleing@25610
  1318
      from MBb have BsubM: "B \<subset># M" by simp
kleing@25610
  1319
      show ?case
kleing@25610
  1320
      proof cases
kleing@25610
  1321
        assume "b=c"
kleing@25610
  1322
        then moreover have "B = C" using MBb MCc by auto
kleing@25610
  1323
        ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
kleing@25610
  1324
      next
kleing@25610
  1325
        assume diff: "b \<noteq> c"
kleing@25610
  1326
        let ?D = "B - {#c#}"
kleing@25610
  1327
        have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
kleing@25610
  1328
          by (auto intro: insert_noteq_member dest: sym)
kleing@25610
  1329
        have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
wenzelm@26145
  1330
        then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
kleing@25610
  1331
        from MBb MCc have "B + {#b#} = C + {#c#}" by blast
wenzelm@26145
  1332
        then have [simp]: "B + {#b#} - {#c#} = C"
kleing@25610
  1333
          using MBb MCc binC cinB by auto
kleing@25610
  1334
        have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
kleing@25610
  1335
          using MBb MCc diff binC cinB
kleing@25610
  1336
          by (auto simp: multiset_add_sub_el_shuffle)
kleing@25759
  1337
        then obtain d where Dfoldd: "fold_msetG f Z ?D d"
kleing@25759
  1338
          using fold_msetG_nonempty by iprover
wenzelm@26145
  1339
        then have "fold_msetG f Z B (f c d)" using cinB
kleing@25759
  1340
          by (rule Diff1_fold_msetG)
wenzelm@26145
  1341
        then have "f c d = u" using IH BsubM Bu by blast
kleing@25610
  1342
        moreover 
kleing@25759
  1343
        have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
kleing@25610
  1344
          by (auto simp: multiset_add_sub_el_shuffle 
kleing@25759
  1345
            dest: fold_msetG.insertI [where x=b])
wenzelm@26145
  1346
        then have "f b d = v" using IH CsubM Cv by blast
kleing@25610
  1347
        ultimately show ?thesis using x\<^isub>1 x\<^isub>2
kleing@25610
  1348
          by (auto simp: left_commute)
kleing@25610
  1349
      qed
kleing@25610
  1350
    qed
kleing@25610
  1351
  qed
kleing@25610
  1352
qed
kleing@25610
  1353
        
wenzelm@26145
  1354
lemma fold_mset_insert_aux:
wenzelm@26145
  1355
  "(fold_msetG f z (A + {#x#}) v) =
kleing@25759
  1356
    (\<exists>y. fold_msetG f z A y \<and> v = f x y)"
nipkow@26178
  1357
apply (rule iffI)
nipkow@26178
  1358
 prefer 2
nipkow@26178
  1359
 apply blast
nipkow@26178
  1360
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
nipkow@26178
  1361
apply (blast intro: fold_msetG_determ)
nipkow@26178
  1362
done
kleing@25610
  1363
wenzelm@26145
  1364
lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
nipkow@26178
  1365
unfolding fold_mset_def by (blast intro: fold_msetG_determ)
kleing@25610
  1366
wenzelm@26145
  1367
lemma fold_mset_insert:
nipkow@26178
  1368
  "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
nipkow@26178
  1369
apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)  
nipkow@26178
  1370
apply (rule the_equality)
nipkow@26178
  1371
 apply (auto cong add: conj_cong 
wenzelm@26145
  1372
     simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
nipkow@26178
  1373
done
kleing@25759
  1374
wenzelm@26145
  1375
lemma fold_mset_insert_idem:
nipkow@26178
  1376
  "fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"
nipkow@26178
  1377
apply (simp add: fold_mset_def fold_mset_insert_aux)
nipkow@26178
  1378
apply (rule the_equality)
nipkow@26178
  1379
 apply (auto cong add: conj_cong 
wenzelm@26145
  1380
     simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
nipkow@26178
  1381
done
kleing@25610
  1382
wenzelm@26145
  1383
lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
nipkow@26178
  1384
by (induct A) (auto simp: fold_mset_insert left_commute [of x])
nipkow@26178
  1385
wenzelm@26145
  1386
lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
nipkow@26178
  1387
using fold_mset_insert [of z "{#}"] by simp
kleing@25610
  1388
wenzelm@26145
  1389
lemma fold_mset_union [simp]:
wenzelm@26145
  1390
  "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
kleing@25759
  1391
proof (induct A)
wenzelm@26145
  1392
  case empty then show ?case by simp
kleing@25759
  1393
next
wenzelm@26145
  1394
  case (add A x)
wenzelm@26145
  1395
  have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
wenzelm@26145
  1396
  then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" 
wenzelm@26145
  1397
    by (simp add: fold_mset_insert)
wenzelm@26145
  1398
  also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
wenzelm@26145
  1399
    by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
wenzelm@26145
  1400
  finally show ?case .
kleing@25759
  1401
qed
kleing@25759
  1402
wenzelm@26145
  1403
lemma fold_mset_fusion:
ballarin@27611
  1404
  assumes "left_commutative g"
ballarin@27611
  1405
  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
  1406
proof -
ballarin@27611
  1407
  interpret left_commutative [g] by fact
ballarin@27611
  1408
  show "PROP ?P" by (induct A) auto
ballarin@27611
  1409
qed
kleing@25610
  1410
wenzelm@26145
  1411
lemma fold_mset_rec:
wenzelm@26145
  1412
  assumes "a \<in># A" 
kleing@25759
  1413
  shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
kleing@25610
  1414
proof -
wenzelm@26145
  1415
  from assms obtain A' where "A = A' + {#a#}"
wenzelm@26145
  1416
    by (blast dest: multi_member_split)
wenzelm@26145
  1417
  then show ?thesis by simp
kleing@25610
  1418
qed
kleing@25610
  1419
wenzelm@26145
  1420
end
wenzelm@26145
  1421
wenzelm@26145
  1422
text {*
wenzelm@26145
  1423
  A note on code generation: When defining some function containing a
wenzelm@26145
  1424
  subterm @{term"fold_mset F"}, code generation is not automatic. When
wenzelm@26145
  1425
  interpreting locale @{text left_commutative} with @{text F}, the
wenzelm@26145
  1426
  would be code thms for @{const fold_mset} become thms like
wenzelm@26145
  1427
  @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
wenzelm@26145
  1428
  contains defined symbols, i.e.\ is not a code thm. Hence a separate
wenzelm@26145
  1429
  constant with its own code thms needs to be introduced for @{text
wenzelm@26145
  1430
  F}. See the image operator below.
wenzelm@26145
  1431
*}
wenzelm@26145
  1432
nipkow@26016
  1433
nipkow@26016
  1434
subsection {* Image *}
nipkow@26016
  1435
haftmann@28708
  1436
definition [code del]:
haftmann@28708
  1437
 "image_mset f = fold_mset (op + o single o f) {#}"
nipkow@26016
  1438
wenzelm@26145
  1439
interpretation image_left_comm: left_commutative ["op + o single o f"]
nipkow@26178
  1440
by (unfold_locales) (simp add:union_ac)
nipkow@26016
  1441
haftmann@28708
  1442
lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
nipkow@26178
  1443
by (simp add: image_mset_def)
nipkow@26016
  1444
haftmann@28708
  1445
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
nipkow@26178
  1446
by (simp add: image_mset_def)
nipkow@26016
  1447
nipkow@26016
  1448
lemma image_mset_insert:
nipkow@26016
  1449
  "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
nipkow@26178
  1450
by (simp add: image_mset_def add_ac)
nipkow@26016
  1451
haftmann@28708
  1452
lemma image_mset_union [simp]:
nipkow@26016
  1453
  "image_mset f (M+N) = image_mset f M + image_mset f N"
nipkow@26178
  1454
apply (induct N)
nipkow@26178
  1455
 apply simp
nipkow@26178
  1456
apply (simp add: union_assoc [symmetric] image_mset_insert)
nipkow@26178
  1457
done
nipkow@26016
  1458
wenzelm@26145
  1459
lemma size_image_mset [simp]: "size (image_mset f M) = size M"
nipkow@26178
  1460
by (induct M) simp_all
nipkow@26016
  1461
wenzelm@26145
  1462
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
nipkow@26178
  1463
by (cases M) auto
nipkow@26016
  1464
wenzelm@26145
  1465
syntax
wenzelm@26145
  1466
  comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
wenzelm@26145
  1467
      ("({#_/. _ :# _#})")
wenzelm@26145
  1468
translations
wenzelm@26145
  1469
  "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
nipkow@26016
  1470
wenzelm@26145
  1471
syntax
wenzelm@26145
  1472
  comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
wenzelm@26145
  1473
      ("({#_/ | _ :# _./ _#})")
nipkow@26016
  1474
translations
nipkow@26033
  1475
  "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
nipkow@26016
  1476
wenzelm@26145
  1477
text {*
wenzelm@26145
  1478
  This allows to write not just filters like @{term "{#x:#M. x<c#}"}
wenzelm@26145
  1479
  but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
wenzelm@26145
  1480
  "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
wenzelm@26145
  1481
  @{term "{#x+x|x:#M. x<c#}"}.
wenzelm@26145
  1482
*}
nipkow@26016
  1483
wenzelm@10249
  1484
end