src/HOL/Library/Multiset.thy
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(*  Title:      HOL/Library/Multiset.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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*)
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header {* Multisets *}
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theory Multiset
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imports 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
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  Mempty :: "'a multiset"  ("{#}") where
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  "{#} = Abs_multiset (\<lambda>a. 0)"
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definition
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  single :: "'a => 'a multiset" where
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  "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
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declare
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  Mempty_def[code func del] single_def[code func del]
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definition
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  count :: "'a multiset => 'a => nat" where
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  "count = Rep_multiset"
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definition
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  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
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  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
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  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
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  union_def[code func del]:
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  "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
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definition
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  diff_def: "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
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definition
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  Zero_multiset_def [simp]: "0 = {#}"
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definition
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  size_def[code func del]: "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)
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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,code func]: "size {#} = 0"
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by (simp add: size_def)
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lemma size_single [simp,code func]: "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,code func]: "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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   333
  "(M + {#a#} = N + {#b#}) =
f9d1bf2fc59c added multiset comprehension
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diff changeset
   334
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
26178
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parents: 26176
diff changeset
   335
by (auto simp add: add_eq_conv_diff)
26016
f9d1bf2fc59c added multiset comprehension
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diff changeset
   336
f9d1bf2fc59c added multiset comprehension
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diff changeset
   337
f9d1bf2fc59c added multiset comprehension
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diff changeset
   338
lemma empty_multiset_count:
f9d1bf2fc59c added multiset comprehension
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parents: 25759
diff changeset
   339
  "(\<forall>x. count A x = 0) = (A = {#})"
26178
nipkow
parents: 26176
diff changeset
   340
by (metis count_empty multiset_eq_conv_count_eq)
26016
f9d1bf2fc59c added multiset comprehension
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diff changeset
   341
f9d1bf2fc59c added multiset comprehension
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diff changeset
   342
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   343
subsubsection {* Intersection *}
3aca7f05cd12 intersection
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   344
3aca7f05cd12 intersection
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   345
lemma multiset_inter_count:
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diff changeset
   346
  "count (A #\<inter> B) x = min (count A x) (count B x)"
nipkow
parents: 26176
diff changeset
   347
by (simp add: multiset_inter_def min_def)
15869
3aca7f05cd12 intersection
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parents: 15867
diff changeset
   348
3aca7f05cd12 intersection
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diff changeset
   349
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
26178
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parents: 26176
diff changeset
   350
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
21214
a91bab12b2bd adjusted two lemma names due to name change in interpretation
haftmann
parents: 20770
diff changeset
   351
    min_max.inf_commute)
15869
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parents: 15867
diff changeset
   352
3aca7f05cd12 intersection
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diff changeset
   353
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
26178
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parents: 26176
diff changeset
   354
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
21214
a91bab12b2bd adjusted two lemma names due to name change in interpretation
haftmann
parents: 20770
diff changeset
   355
    min_max.inf_assoc)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   356
3aca7f05cd12 intersection
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parents: 15867
diff changeset
   357
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
26178
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parents: 26176
diff changeset
   358
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
15869
3aca7f05cd12 intersection
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parents: 15867
diff changeset
   359
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   360
lemmas multiset_inter_ac =
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   361
  multiset_inter_commute
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   362
  multiset_inter_assoc
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   363
  multiset_inter_left_commute
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   364
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   365
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
26178
nipkow
parents: 26176
diff changeset
   366
by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   367
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   368
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
26178
nipkow
parents: 26176
diff changeset
   369
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   370
    split: split_if_asm)
26178
nipkow
parents: 26176
diff changeset
   371
apply clarsimp
nipkow
parents: 26176
diff changeset
   372
apply (erule_tac x = a in allE)
nipkow
parents: 26176
diff changeset
   373
apply auto
nipkow
parents: 26176
diff changeset
   374
done
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   375
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   376
26016
f9d1bf2fc59c added multiset comprehension
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parents: 25759
diff changeset
   377
subsubsection {* Comprehension (filter) *}
f9d1bf2fc59c added multiset comprehension
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parents: 25759
diff changeset
   378
f9d1bf2fc59c added multiset comprehension
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parents: 25759
diff changeset
   379
lemma MCollect_empty[simp, code func]: "MCollect {#} P = {#}"
26178
nipkow
parents: 26176
diff changeset
   380
by (simp add: MCollect_def Mempty_def Abs_multiset_inject
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   381
    in_multiset expand_fun_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   382
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   383
lemma MCollect_single[simp, code func]:
26178
nipkow
parents: 26176
diff changeset
   384
  "MCollect {#x#} P = (if P x then {#x#} else {#})"
nipkow
parents: 26176
diff changeset
   385
by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   386
    in_multiset expand_fun_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   387
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   388
lemma MCollect_union[simp, code func]:
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   389
  "MCollect (M+N) f = MCollect M f + MCollect N f"
26178
nipkow
parents: 26176
diff changeset
   390
by (simp add: MCollect_def union_def Abs_multiset_inject
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   391
    in_multiset expand_fun_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   392
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   393
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   394
subsection {* Induction and case splits *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   395
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   396
lemma setsum_decr:
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   397
  "finite F ==> (0::nat) < f a ==>
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   398
    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
26178
nipkow
parents: 26176
diff changeset
   399
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   400
 apply auto
nipkow
parents: 26176
diff changeset
   401
apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow
parents: 26176
diff changeset
   402
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   403
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   404
lemma rep_multiset_induct_aux:
26178
nipkow
parents: 26176
diff changeset
   405
assumes 1: "P (\<lambda>a. (0::nat))"
nipkow
parents: 26176
diff changeset
   406
  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow
parents: 26176
diff changeset
   407
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow
parents: 26176
diff changeset
   408
apply (unfold multiset_def)
nipkow
parents: 26176
diff changeset
   409
apply (induct_tac n, simp, clarify)
nipkow
parents: 26176
diff changeset
   410
 apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow
parents: 26176
diff changeset
   411
  apply simp
nipkow
parents: 26176
diff changeset
   412
  apply (rule 1)
nipkow
parents: 26176
diff changeset
   413
 apply (rule ext, force, clarify)
nipkow
parents: 26176
diff changeset
   414
apply (frule setsum_SucD, clarify)
nipkow
parents: 26176
diff changeset
   415
apply (rename_tac a)
nipkow
parents: 26176
diff changeset
   416
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow
parents: 26176
diff changeset
   417
 prefer 2
nipkow
parents: 26176
diff changeset
   418
 apply (rule finite_subset)
nipkow
parents: 26176
diff changeset
   419
  prefer 2
nipkow
parents: 26176
diff changeset
   420
  apply assumption
nipkow
parents: 26176
diff changeset
   421
 apply simp
nipkow
parents: 26176
diff changeset
   422
 apply blast
nipkow
parents: 26176
diff changeset
   423
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow
parents: 26176
diff changeset
   424
 prefer 2
nipkow
parents: 26176
diff changeset
   425
 apply (rule ext)
nipkow
parents: 26176
diff changeset
   426
 apply (simp (no_asm_simp))
nipkow
parents: 26176
diff changeset
   427
 apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow
parents: 26176
diff changeset
   428
apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow
parents: 26176
diff changeset
   429
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow
parents: 26176
diff changeset
   430
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow
parents: 26176
diff changeset
   431
 prefer 2
nipkow
parents: 26176
diff changeset
   432
 apply blast
nipkow
parents: 26176
diff changeset
   433
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow
parents: 26176
diff changeset
   434
 prefer 2
nipkow
parents: 26176
diff changeset
   435
 apply blast
nipkow
parents: 26176
diff changeset
   436
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow
parents: 26176
diff changeset
   437
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   438
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   439
theorem rep_multiset_induct:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   440
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   441
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
26178
nipkow
parents: 26176
diff changeset
   442
using rep_multiset_induct_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   443
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   444
theorem multiset_induct [case_names empty add, induct type: multiset]:
26178
nipkow
parents: 26176
diff changeset
   445
assumes empty: "P {#}"
nipkow
parents: 26176
diff changeset
   446
  and add: "!!M x. P M ==> P (M + {#x#})"
nipkow
parents: 26176
diff changeset
   447
shows "P M"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   448
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   449
  note defns = union_def single_def Mempty_def
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   450
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   451
    apply (rule Rep_multiset_inverse [THEN subst])
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   452
    apply (rule Rep_multiset [THEN rep_multiset_induct])
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   453
     apply (rule empty [unfolded defns])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   454
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   455
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   456
     apply (simp add: expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   457
    apply (erule ssubst)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   458
    apply (erule Abs_multiset_inverse [THEN subst])
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   459
    apply (drule add [unfolded defns, simplified])
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   460
    apply(simp add:in_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   461
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   462
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   463
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   464
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   465
by (induct M) auto
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   466
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   467
lemma multiset_cases [cases type, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
   468
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   469
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   470
shows "P"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   471
proof (cases "M = {#}")
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   472
  assume "M = {#}" then show ?thesis using em by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   473
next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   474
  assume "M \<noteq> {#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   475
  then obtain M' m where "M = M' + {#m#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   476
    by (blast dest: multi_nonempty_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   477
  then show ?thesis using add by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   478
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   479
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   480
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
26178
nipkow
parents: 26176
diff changeset
   481
apply (cases M)
nipkow
parents: 26176
diff changeset
   482
 apply simp
nipkow
parents: 26176
diff changeset
   483
apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow
parents: 26176
diff changeset
   484
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   485
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   486
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
26178
nipkow
parents: 26176
diff changeset
   487
apply (subst multiset_eq_conv_count_eq)
nipkow
parents: 26176
diff changeset
   488
apply auto
nipkow
parents: 26176
diff changeset
   489
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   490
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   491
declare multiset_typedef [simp del]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   492
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   493
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
26178
nipkow
parents: 26176
diff changeset
   494
by (cases "B = {#}") (auto dest: multi_member_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   495
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   496
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   497
subsection {* Orderings *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   498
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   499
subsubsection {* Well-foundedness *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   500
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   501
definition
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   502
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   503
  "mult1 r =
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   504
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   505
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   506
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21214
diff changeset
   507
definition
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   508
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   509
  "mult r = (mult1 r)\<^sup>+"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   510
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   511
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
26178
nipkow
parents: 26176
diff changeset
   512
by (simp add: mult1_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   513
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   514
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   515
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   516
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
19582
a669c98b9c24 get rid of 'concl is';
wenzelm
parents: 19564
diff changeset
   517
  (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   518
proof (unfold mult1_def)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   519
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   520
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   521
  let ?case1 = "?case1 {(N, M). ?R N M}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   522
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   523
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   524
  then have "\<exists>a' M0' K.
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   525
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   526
  then show "?case1 \<or> ?case2"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   527
  proof (elim exE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   528
    fix a' M0' K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   529
    assume N: "N = M0' + K" and r: "?r K a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   530
    assume "M0 + {#a#} = M0' + {#a'#}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   531
    then have "M0 = M0' \<and> a = a' \<or>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   532
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   533
      by (simp only: add_eq_conv_ex)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   534
    then show ?thesis
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   535
    proof (elim disjE conjE exE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   536
      assume "M0 = M0'" "a = a'"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   537
      with N r have "?r K a \<and> N = M0 + K" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   538
      then have ?case2 .. then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   539
    next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   540
      fix K'
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   541
      assume "M0' = K' + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   542
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   543
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   544
      assume "M0 = K' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   545
      with r have "?R (K' + K) M0" by blast
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   546
      with n have ?case1 by simp then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   547
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   548
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   549
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   550
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   551
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   552
proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   553
  let ?R = "mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   554
  let ?W = "acc ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   555
  {
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   556
    fix M M0 a
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   557
    assume M0: "M0 \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   558
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   559
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   560
    have "M0 + {#a#} \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   561
    proof (rule accI [of "M0 + {#a#}"])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   562
      fix N
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   563
      assume "(N, M0 + {#a#}) \<in> ?R"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   564
      then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   565
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   566
        by (rule less_add)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   567
      then show "N \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   568
      proof (elim exE disjE conjE)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   569
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   570
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   571
        from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   572
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   573
      next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   574
        fix K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   575
        assume N: "N = M0 + K"
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   576
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   577
        then have "M0 + K \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   578
        proof (induct K)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   579
          case empty
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   580
          from M0 show "M0 + {#} \<in> ?W" by simp
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   581
        next
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   582
          case (add K x)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   583
          from add.prems have "(x, a) \<in> r" by simp
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   584
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   585
          moreover from add have "M0 + K \<in> ?W" by simp
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   586
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   587
          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   588
        qed
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   589
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   590
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   591
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   592
  } note tedious_reasoning = this
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   593
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   594
  assume wf: "wf r"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   595
  fix M
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   596
  show "M \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   597
  proof (induct M)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   598
    show "{#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   599
    proof (rule accI)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   600
      fix b assume "(b, {#}) \<in> ?R"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   601
      with not_less_empty show "b \<in> ?W" by contradiction
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   602
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   603
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   604
    fix M a assume "M \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   605
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   606
    proof induct
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   607
      fix a
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   608
      assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   609
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   610
      proof
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   611
        fix M assume "M \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   612
        then show "M + {#a#} \<in> ?W"
23373
ead82c82da9e tuned proofs: avoid implicit prems;
wenzelm
parents: 23281
diff changeset
   613
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   614
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   615
    qed
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   616
    from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   617
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   618
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   619
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   620
theorem wf_mult1: "wf r ==> wf (mult1 r)"
26178
nipkow
parents: 26176
diff changeset
   621
by (rule acc_wfI) (rule all_accessible)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   622
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   623
theorem wf_mult: "wf r ==> wf (mult r)"
26178
nipkow
parents: 26176
diff changeset
   624
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   625
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   626
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   627
subsubsection {* Closure-free presentation *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   628
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   629
(*Badly needed: a linear arithmetic procedure for multisets*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   630
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   631
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
26178
nipkow
parents: 26176
diff changeset
   632
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   633
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   634
text {* One direction. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   635
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   636
lemma mult_implies_one_step:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   637
  "trans r ==> (M, N) \<in> mult r ==>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   638
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   639
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
26178
nipkow
parents: 26176
diff changeset
   640
apply (unfold mult_def mult1_def set_of_def)
nipkow
parents: 26176
diff changeset
   641
apply (erule converse_trancl_induct, clarify)
nipkow
parents: 26176
diff changeset
   642
 apply (rule_tac x = M0 in exI, simp, clarify)
nipkow
parents: 26176
diff changeset
   643
apply (case_tac "a :# K")
nipkow
parents: 26176
diff changeset
   644
 apply (rule_tac x = I in exI)
nipkow
parents: 26176
diff changeset
   645
 apply (simp (no_asm))
nipkow
parents: 26176
diff changeset
   646
 apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
nipkow
parents: 26176
diff changeset
   647
 apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow
parents: 26176
diff changeset
   648
 apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow
parents: 26176
diff changeset
   649
 apply (simp add: diff_union_single_conv)
nipkow
parents: 26176
diff changeset
   650
 apply (simp (no_asm_use) add: trans_def)
nipkow
parents: 26176
diff changeset
   651
 apply blast
nipkow
parents: 26176
diff changeset
   652
apply (subgoal_tac "a :# I")
nipkow
parents: 26176
diff changeset
   653
 apply (rule_tac x = "I - {#a#}" in exI)
nipkow
parents: 26176
diff changeset
   654
 apply (rule_tac x = "J + {#a#}" in exI)
nipkow
parents: 26176
diff changeset
   655
 apply (rule_tac x = "K + Ka" in exI)
nipkow
parents: 26176
diff changeset
   656
 apply (rule conjI)
nipkow
parents: 26176
diff changeset
   657
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow
parents: 26176
diff changeset
   658
 apply (rule conjI)
nipkow
parents: 26176
diff changeset
   659
  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow
parents: 26176
diff changeset
   660
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow
parents: 26176
diff changeset
   661
 apply (simp (no_asm_use) add: trans_def)
nipkow
parents: 26176
diff changeset
   662
 apply blast
nipkow
parents: 26176
diff changeset
   663
apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow
parents: 26176
diff changeset
   664
 apply simp
nipkow
parents: 26176
diff changeset
   665
apply (simp (no_asm))
nipkow
parents: 26176
diff changeset
   666
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   667
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   668
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   669
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   670
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   671
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   672
apply (erule size_eq_Suc_imp_elem [THEN exE])
nipkow
parents: 26176
diff changeset
   673
apply (drule elem_imp_eq_diff_union, auto)
nipkow
parents: 26176
diff changeset
   674
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   675
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   676
lemma one_step_implies_mult_aux:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   677
  "trans r ==>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   678
    \<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))
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   679
      --> (I + K, I + J) \<in> mult r"
26178
nipkow
parents: 26176
diff changeset
   680
apply (induct_tac n, auto)
nipkow
parents: 26176
diff changeset
   681
apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow
parents: 26176
diff changeset
   682
apply (rename_tac "J'", simp)
nipkow
parents: 26176
diff changeset
   683
apply (erule notE, auto)
nipkow
parents: 26176
diff changeset
   684
apply (case_tac "J' = {#}")
nipkow
parents: 26176
diff changeset
   685
 apply (simp add: mult_def)
nipkow
parents: 26176
diff changeset
   686
 apply (rule r_into_trancl)
nipkow
parents: 26176
diff changeset
   687
 apply (simp add: mult1_def set_of_def, blast)
nipkow
parents: 26176
diff changeset
   688
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow
parents: 26176
diff changeset
   689
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow
parents: 26176
diff changeset
   690
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
nipkow
parents: 26176
diff changeset
   691
apply (erule ssubst)
nipkow
parents: 26176
diff changeset
   692
apply (simp add: Ball_def, auto)
nipkow
parents: 26176
diff changeset
   693
apply (subgoal_tac
nipkow
parents: 26176
diff changeset
   694
  "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
nipkow
parents: 26176
diff changeset
   695
    (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow
parents: 26176
diff changeset
   696
 prefer 2
nipkow
parents: 26176
diff changeset
   697
 apply force
nipkow
parents: 26176
diff changeset
   698
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
nipkow
parents: 26176
diff changeset
   699
apply (erule trancl_trans)
nipkow
parents: 26176
diff changeset
   700
apply (rule r_into_trancl)
nipkow
parents: 26176
diff changeset
   701
apply (simp add: mult1_def set_of_def)
nipkow
parents: 26176
diff changeset
   702
apply (rule_tac x = a in exI)
nipkow
parents: 26176
diff changeset
   703
apply (rule_tac x = "I + J'" in exI)
nipkow
parents: 26176
diff changeset
   704
apply (simp add: union_ac)
nipkow
parents: 26176
diff changeset
   705
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   706
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   707
lemma one_step_implies_mult:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   708
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   709
    ==> (I + K, I + J) \<in> mult r"
26178
nipkow
parents: 26176
diff changeset
   710
using one_step_implies_mult_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   711
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   712
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   713
subsubsection {* Partial-order properties *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   714
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 11868
diff changeset
   715
instance multiset :: (type) ord ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   716
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   717
defs (overloaded)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   718
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   719
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   720
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   721
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
26178
nipkow
parents: 26176
diff changeset
   722
unfolding trans_def by (blast intro: order_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   723
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   724
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   725
 \medskip Irreflexivity.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   726
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   727
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   728
lemma mult_irrefl_aux:
26178
nipkow
parents: 26176
diff changeset
   729
  "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
nipkow
parents: 26176
diff changeset
   730
by (induct rule: finite_induct) (auto intro: order_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   731
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   732
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   733
apply (unfold less_multiset_def, auto)
nipkow
parents: 26176
diff changeset
   734
apply (drule trans_base_order [THEN mult_implies_one_step], auto)
nipkow
parents: 26176
diff changeset
   735
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
nipkow
parents: 26176
diff changeset
   736
apply (simp add: set_of_eq_empty_iff)
nipkow
parents: 26176
diff changeset
   737
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   738
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   739
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
26178
nipkow
parents: 26176
diff changeset
   740
using insert mult_less_not_refl by fast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   741
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   742
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   743
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   744
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   745
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   746
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   747
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   748
text {* Asymmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   749
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   750
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   751
apply auto
nipkow
parents: 26176
diff changeset
   752
apply (rule mult_less_not_refl [THEN notE])
nipkow
parents: 26176
diff changeset
   753
apply (erule mult_less_trans, assumption)
nipkow
parents: 26176
diff changeset
   754
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   755
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   756
theorem mult_less_asym:
26178
nipkow
parents: 26176
diff changeset
   757
  "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
nipkow
parents: 26176
diff changeset
   758
using mult_less_not_sym by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   759
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   760
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   761
unfolding le_multiset_def by auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   762
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   763
text {* Anti-symmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   764
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   765
theorem mult_le_antisym:
26178
nipkow
parents: 26176
diff changeset
   766
  "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   767
unfolding le_multiset_def by (blast dest: mult_less_not_sym)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   768
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   769
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   770
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   771
theorem mult_le_trans:
26178
nipkow
parents: 26176
diff changeset
   772
  "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   773
unfolding le_multiset_def by (blast intro: mult_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   774
11655
923e4d0d36d5 tuned parentheses in relational expressions;
wenzelm
parents: 11549
diff changeset
   775
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
26178
nipkow
parents: 26176
diff changeset
   776
unfolding le_multiset_def by auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   777
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   778
text {* Partial order. *}
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   779
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   780
instance multiset :: (order) order
26178
nipkow
parents: 26176
diff changeset
   781
apply intro_classes
nipkow
parents: 26176
diff changeset
   782
   apply (rule mult_less_le)
nipkow
parents: 26176
diff changeset
   783
  apply (rule mult_le_refl)
nipkow
parents: 26176
diff changeset
   784
 apply (erule mult_le_trans, assumption)
nipkow
parents: 26176
diff changeset
   785
apply (erule mult_le_antisym, assumption)
nipkow
parents: 26176
diff changeset
   786
done
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   787
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   788
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   789
subsubsection {* Monotonicity of multiset union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   790
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   791
lemma mult1_union:
26178
nipkow
parents: 26176
diff changeset
   792
  "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
nipkow
parents: 26176
diff changeset
   793
apply (unfold mult1_def)
nipkow
parents: 26176
diff changeset
   794
apply auto
nipkow
parents: 26176
diff changeset
   795
apply (rule_tac x = a in exI)
nipkow
parents: 26176
diff changeset
   796
apply (rule_tac x = "C + M0" in exI)
nipkow
parents: 26176
diff changeset
   797
apply (simp add: union_assoc)
nipkow
parents: 26176
diff changeset
   798
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   799
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   800
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   801
apply (unfold less_multiset_def mult_def)
nipkow
parents: 26176
diff changeset
   802
apply (erule trancl_induct)
nipkow
parents: 26176
diff changeset
   803
 apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
nipkow
parents: 26176
diff changeset
   804
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
nipkow
parents: 26176
diff changeset
   805
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   806
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   807
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   808
apply (subst union_commute [of B C])
nipkow
parents: 26176
diff changeset
   809
apply (subst union_commute [of D C])
nipkow
parents: 26176
diff changeset
   810
apply (erule union_less_mono2)
nipkow
parents: 26176
diff changeset
   811
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   812
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   813
lemma union_less_mono:
26178
nipkow
parents: 26176
diff changeset
   814
  "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   815
by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   816
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   817
lemma union_le_mono:
26178
nipkow
parents: 26176
diff changeset
   818
  "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   819
unfolding le_multiset_def
nipkow
parents: 26176
diff changeset
   820
by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   821
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   822
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   823
apply (unfold le_multiset_def less_multiset_def)
nipkow
parents: 26176
diff changeset
   824
apply (case_tac "M = {#}")
nipkow
parents: 26176
diff changeset
   825
 prefer 2
nipkow
parents: 26176
diff changeset
   826
 apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
nipkow
parents: 26176
diff changeset
   827
  prefer 2
nipkow
parents: 26176
diff changeset
   828
  apply (rule one_step_implies_mult)
nipkow
parents: 26176
diff changeset
   829
    apply (simp only: trans_def)
nipkow
parents: 26176
diff changeset
   830
    apply auto
nipkow
parents: 26176
diff changeset
   831
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   832
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   833
lemma union_upper1: "A <= A + (B::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   834
proof -
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   835
  have "A + {#} <= A + B" by (blast intro: union_le_mono)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   836
  then show ?thesis by simp
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   837
qed
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   838
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   839
lemma union_upper2: "B <= A + (B::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   840
by (subst union_commute) (rule union_upper1)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   841
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   842
instance multiset :: (order) pordered_ab_semigroup_add
26178
nipkow
parents: 26176
diff changeset
   843
apply intro_classes
nipkow
parents: 26176
diff changeset
   844
apply (erule union_le_mono[OF mult_le_refl])
nipkow
parents: 26176
diff changeset
   845
done
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   846
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   847
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   848
subsection {* Link with lists *}
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   849
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   850
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   851
  "multiset_of [] = {#}" |
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   852
  "multiset_of (a # x) = multiset_of x + {# a #}"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   853
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   854
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
26178
nipkow
parents: 26176
diff changeset
   855
by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   856
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   857
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
26178
nipkow
parents: 26176
diff changeset
   858
by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   859
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   860
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
26178
nipkow
parents: 26176
diff changeset
   861
by (induct x) auto
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   862
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   863
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
26178
nipkow
parents: 26176
diff changeset
   864
by (induct xs) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   865
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   866
lemma multiset_of_append [simp]:
26178
nipkow
parents: 26176
diff changeset
   867
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
nipkow
parents: 26176
diff changeset
   868
by (induct xs arbitrary: ys) (auto simp: union_ac)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   869
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   870
lemma surj_multiset_of: "surj multiset_of"
26178
nipkow
parents: 26176
diff changeset
   871
apply (unfold surj_def)
nipkow
parents: 26176
diff changeset
   872
apply (rule allI)
nipkow
parents: 26176
diff changeset
   873
apply (rule_tac M = y in multiset_induct)
nipkow
parents: 26176
diff changeset
   874
 apply auto
nipkow
parents: 26176
diff changeset
   875
apply (rule_tac x = "x # xa" in exI)
nipkow
parents: 26176
diff changeset
   876
apply auto
nipkow
parents: 26176
diff changeset
   877
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   878
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25140
diff changeset
   879
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
26178
nipkow
parents: 26176
diff changeset
   880
by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   881
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   882
lemma distinct_count_atmost_1:
26178
nipkow
parents: 26176
diff changeset
   883
  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
nipkow
parents: 26176
diff changeset
   884
apply (induct x, simp, rule iffI, simp_all)
nipkow
parents: 26176
diff changeset
   885
apply (rule conjI)
nipkow
parents: 26176
diff changeset
   886
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
nipkow
parents: 26176
diff changeset
   887
apply (erule_tac x = a in allE, simp, clarify)
nipkow
parents: 26176
diff changeset
   888
apply (erule_tac x = aa in allE, simp)
nipkow
parents: 26176
diff changeset
   889
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   890
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   891
lemma multiset_of_eq_setD:
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   892
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
26178
nipkow
parents: 26176
diff changeset
   893
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   894
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   895
lemma set_eq_iff_multiset_of_eq_distinct:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   896
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   897
    (set x = set y) = (multiset_of x = multiset_of y)"
26178
nipkow
parents: 26176
diff changeset
   898
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   899
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   900
lemma set_eq_iff_multiset_of_remdups_eq:
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   901
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
26178
nipkow
parents: 26176
diff changeset
   902
apply (rule iffI)
nipkow
parents: 26176
diff changeset
   903
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
nipkow
parents: 26176
diff changeset
   904
apply (drule distinct_remdups [THEN distinct_remdups
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   905
      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
26178
nipkow
parents: 26176
diff changeset
   906
apply simp
nipkow
parents: 26176
diff changeset
   907
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   908
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   909
lemma multiset_of_compl_union [simp]:
26178
nipkow
parents: 26176
diff changeset
   910
  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
nipkow
parents: 26176
diff changeset
   911
by (induct xs) (auto simp: union_ac)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   912
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   913
lemma count_filter:
26178
nipkow
parents: 26176
diff changeset
   914
  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
nipkow
parents: 26176
diff changeset
   915
by (induct xs) auto
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   916
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   917
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
26178
nipkow
parents: 26176
diff changeset
   918
apply (induct ls arbitrary: i)
nipkow
parents: 26176
diff changeset
   919
 apply simp
nipkow
parents: 26176
diff changeset
   920
apply (case_tac i)
nipkow
parents: 26176
diff changeset
   921
 apply auto
nipkow
parents: 26176
diff changeset
   922
done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   923
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   924
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
26178
nipkow
parents: 26176
diff changeset
   925
by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   926
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   927
lemma multiset_of_eq_length:
26178
nipkow
parents: 26176
diff changeset
   928
assumes "multiset_of xs = multiset_of ys"
nipkow
parents: 26176
diff changeset
   929
shows "length xs = length ys"
nipkow
parents: 26176
diff changeset
   930
using assms
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   931
proof (induct arbitrary: ys rule: length_induct)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   932
  case (1 xs ys)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   933
  show ?case
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   934
  proof (cases xs)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   935
    case Nil with "1.prems" show ?thesis by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   936
  next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   937
    case (Cons x xs')
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   938
    note xCons = Cons
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   939
    show ?thesis
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   940
    proof (cases ys)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   941
      case Nil
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   942
      with "1.prems" Cons show ?thesis by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   943
    next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   944
      case (Cons y ys')
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   945
      have x_in_ys: "x = y \<or> x \<in> set ys'"
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   946
      proof (cases "x = y")
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   947
	case True then show ?thesis ..
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   948
      next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   949
	case False
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   950
	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   951
	with False show ?thesis by (simp add: mem_set_multiset_eq)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   952
      qed
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   953
      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   954
	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   955
      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   956
	apply -
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   957
	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   958
	apply fastsimp
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   959
	done
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   960
      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   961
      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   962
      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   963
    qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   964
  qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   965
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   966
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   967
text {*
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   968
  This lemma shows which properties suffice to show that a function
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   969
  @{text "f"} with @{text "f xs = ys"} behaves like sort.
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   970
*}
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   971
lemma properties_for_sort:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   972
  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   973
proof (induct xs arbitrary: ys)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   974
  case Nil then show ?case by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   975
next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   976
  case (Cons x xs)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   977
  then have "x \<in> set ys"
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   978
    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   979
  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   980
    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   981
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   982
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   983
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   984
subsection {* Pointwise ordering induced by count *}
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   985
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   986
definition
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   987
  mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   988
  "(A \<le># B) = (\<forall>a. count A a \<le> count B a)"
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   989
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   990
definition
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   991
  mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   992
  "(A <# B) = (A \<le># B \<and> A \<noteq> B)"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   993
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   994
notation mset_le  (infix "\<subseteq>#" 50)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   995
notation mset_less  (infix "\<subset>#" 50)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   996
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   997
lemma mset_le_refl[simp]: "A \<le># A"
26178
nipkow
parents: 26176
diff changeset
   998
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   999
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1000
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
26178
nipkow
parents: 26176
diff changeset
  1001
unfolding mset_le_def by (fast intro: order_trans)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1002
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1003
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
26178
nipkow
parents: 26176
diff changeset
  1004
apply (unfold mset_le_def)
nipkow
parents: 26176
diff changeset
  1005
apply (rule multiset_eq_conv_count_eq [THEN iffD2])
nipkow
parents: 26176
diff changeset
  1006
apply (blast intro: order_antisym)
nipkow
parents: 26176
diff changeset
  1007
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1008
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1009
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
26178
nipkow
parents: 26176
diff changeset
  1010
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
nipkow
parents: 26176
diff changeset
  1011
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
nipkow
parents: 26176
diff changeset
  1012
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1013
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1014
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
26178
nipkow
parents: 26176
diff changeset
  1015
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1016
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1017
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
26178
nipkow
parents: 26176
diff changeset
  1018
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1019
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1020
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
26178
nipkow
parents: 26176
diff changeset
  1021
apply (unfold mset_le_def)
nipkow
parents: 26176
diff changeset
  1022
apply auto
nipkow
parents: 26176
diff changeset
  1023
apply (erule_tac x = a in allE)+
nipkow
parents: 26176
diff changeset
  1024
apply auto
nipkow
parents: 26176
diff changeset
  1025
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1026
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1027
lemma mset_le_add_left[simp]: "A \<le># A + B"
26178
nipkow
parents: 26176
diff changeset
  1028
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1029
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1030
lemma mset_le_add_right[simp]: "B \<le># A + B"
26178
nipkow
parents: 26176
diff changeset
  1031
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1032
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1033
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
26178
nipkow
parents: 26176
diff changeset
  1034
by (simp add: mset_le_def)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1035
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1036
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
26178
nipkow
parents: 26176
diff changeset
  1037
by (simp add: multiset_eq_conv_count_eq mset_le_def)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1038
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1039
lemma mset_le_multiset_union_diff_commute:
26178
nipkow
parents: 26176
diff changeset
  1040
assumes "B \<le># A"
nipkow
parents: 26176
diff changeset
  1041
shows "A - B + C = A + C - B"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1042
proof -
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1043
  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1044
  from this obtain D where "A = B + D" ..
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1045
  then show ?thesis
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1046
    apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1047
    apply (subst union_commute)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1048
    apply (subst multiset_diff_union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1049
    apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1050
    apply (simp add: diff_cancel)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1051
    apply (subst union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1052
    apply (subst union_commute[of "B" _])
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1053
    apply (subst multiset_diff_union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1054
    apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1055
    apply (simp add: diff_cancel)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1056
    done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1057
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1058
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1059
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
26178
nipkow
parents: 26176
diff changeset
  1060
apply (induct xs)
nipkow
parents: 26176
diff changeset
  1061
 apply auto
nipkow
parents: 26176
diff changeset
  1062
apply (rule mset_le_trans)
nipkow
parents: 26176
diff changeset
  1063
 apply auto
nipkow
parents: 26176
diff changeset
  1064
done
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1065
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1066
lemma multiset_of_update:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1067
  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1068
proof (induct ls arbitrary: i)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1069
  case Nil then show ?case by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1070
next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1071
  case (Cons x xs)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1072
  show ?case
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1073
  proof (cases i)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1074
    case 0 then show ?thesis by simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1075
  next
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1076
    case (Suc i')
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1077
    with Cons show ?thesis
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1078
      apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1079
      apply (subst union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1080
      apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1081
      apply (subst union_assoc [symmetric])
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1082
      apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1083
      apply (rule mset_le_multiset_union_diff_commute)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1084
      apply (simp add: mset_le_single nth_mem_multiset_of)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1085
      done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1086
  qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1087
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1088
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1089
lemma multiset_of_swap:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1090
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1091
    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
26178
nipkow
parents: 26176
diff changeset
  1092
apply (case_tac "i = j")
nipkow
parents: 26176
diff changeset
  1093
 apply simp
nipkow
parents: 26176
diff changeset
  1094
apply (simp add: multiset_of_update)
nipkow
parents: 26176
diff changeset
  1095
apply (subst elem_imp_eq_diff_union[symmetric])
nipkow
parents: 26176
diff changeset
  1096
 apply (simp add: nth_mem_multiset_of)
nipkow
parents: 26176
diff changeset
  1097
apply simp
nipkow
parents: 26176
diff changeset
  1098
done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1099
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1100
interpretation mset_order: order ["op \<le>#" "op <#"]
26178
nipkow
parents: 26176
diff changeset
  1101
by (auto intro: order.intro mset_le_refl mset_le_antisym
25208
1a7318a04068 changed order of class parameters
haftmann
parents: 25162
diff changeset
  1102
    mset_le_trans simp: mset_less_def)
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1103
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1104
interpretation mset_order_cancel_semigroup:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1105
    pordered_cancel_ab_semigroup_add ["op +" "op \<le>#" "op <#"]
26178
nipkow
parents: 26176
diff changeset
  1106
by unfold_locales (erule mset_le_mono_add [OF mset_le_refl])
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1107
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1108
interpretation mset_order_semigroup_cancel:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1109
    pordered_ab_semigroup_add_imp_le ["op +" "op \<le>#" "op <#"]
26178
nipkow
parents: 26176
diff changeset
  1110
by (unfold_locales) simp
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1111
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1112
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1113
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
26178
nipkow
parents: 26176
diff changeset
  1114
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow
parents: 26176
diff changeset
  1115
apply (erule_tac x=x in allE)
nipkow
parents: 26176
diff changeset
  1116
apply auto
nipkow
parents: 26176
diff changeset
  1117
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1118
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1119
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
26178
nipkow
parents: 26176
diff changeset
  1120
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow
parents: 26176
diff changeset
  1121
apply (erule_tac x = x in allE)
nipkow
parents: 26176
diff changeset
  1122
apply auto
nipkow
parents: 26176
diff changeset
  1123
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1124
  
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1125
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
26178
nipkow
parents: 26176
diff changeset
  1126
apply (rule conjI)
nipkow
parents: 26176
diff changeset
  1127
 apply (simp add: mset_lessD)
nipkow
parents: 26176
diff changeset
  1128
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow
parents: 26176
diff changeset
  1129
apply safe
nipkow
parents: 26176
diff changeset
  1130
 apply (erule_tac x = a in allE)
nipkow
parents: 26176
diff changeset
  1131
 apply (auto split: split_if_asm)
nipkow
parents: 26176
diff changeset
  1132
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1133
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1134
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
26178
nipkow
parents: 26176
diff changeset
  1135
apply (rule conjI)
nipkow
parents: 26176
diff changeset
  1136
 apply (simp add: mset_leD)
nipkow
parents: 26176
diff changeset
  1137
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
nipkow
parents: 26176
diff changeset
  1138
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1139
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1140
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" 
26178
nipkow
parents: 26176
diff changeset
  1141
by (induct A) (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1142
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1143
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
26178
nipkow
parents: 26176
diff changeset
  1144
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1145
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1146
lemma multi_psub_self[simp]: "A \<subset># A = False"
26178
nipkow
parents: 26176
diff changeset
  1147
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1148
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1149
lemma mset_less_add_bothsides:
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1150
  "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
26178
nipkow
parents: 26176
diff changeset
  1151
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1152
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1153
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
26178
nipkow
parents: 26176
diff changeset
  1154
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1155
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1156
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1157
proof (induct A arbitrary: B)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1158
  case (empty M)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1159
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1160
  then obtain M' x where "M = M' + {#x#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1161
    by (blast dest: multi_nonempty_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1162
  then show ?case by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1163
next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1164
  case (add S x T)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1165
  have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1166
  have SxsubT: "S + {#x#} \<subset># T" by fact
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1167
  then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1168
  then obtain T' where T: "T = T' + {#x#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1169
    by (blast dest: multi_member_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1170
  then have "S \<subset># T'" using SxsubT 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1171
    by (blast intro: mset_less_add_bothsides)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1172
  then have "size S < size T'" using IH by simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1173
  then show ?case using T by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1174
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1175
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1176
lemmas mset_less_trans = mset_order.less_eq_less.less_trans
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1177
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1178
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
26178
nipkow
parents: 26176
diff changeset
  1179
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1180
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1181
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1182
subsection {* Strong induction and subset induction for multisets *}
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1183
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
  1184
text {* Well-foundedness of proper subset operator: *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1185
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1186
text {* proper multiset subset *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1187
definition
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1188
  mset_less_rel :: "('a multiset * 'a multiset) set" where
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1189
  "mset_less_rel = {(A,B). A \<subset># B}"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1190
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1191
lemma multiset_add_sub_el_shuffle: 
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1192
  assumes "c \<in># B" and "b \<noteq> c" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1193
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1194
proof -
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1195
  from `c \<in># B` obtain A where B: "B = A + {#c#}" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1196
    by (blast dest: multi_member_split)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1197
  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1198
  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1199
    by (simp add: union_ac)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1200
  then show ?thesis using B by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1201
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1202
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1203
lemma wf_mset_less_rel: "wf mset_less_rel"
26178
nipkow
parents: 26176
diff changeset
  1204
apply (unfold mset_less_rel_def)
nipkow
parents: 26176
diff changeset
  1205
apply (rule wf_measure [THEN wf_subset, where f1=size])
nipkow
parents: 26176
diff changeset
  1206
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
nipkow
parents: 26176
diff changeset
  1207
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1208
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
  1209
text {* The induction rules: *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1210
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1211
lemma full_multiset_induct [case_names less]:
26178
nipkow
parents: 26176
diff changeset
  1212
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
nipkow
parents: 26176
diff changeset
  1213
shows "P B"
nipkow
parents: 26176
diff changeset
  1214
apply (rule wf_mset_less_rel [THEN wf_induct])
nipkow
parents: 26176
diff changeset
  1215
apply (rule ih, auto simp: mset_less_rel_def)
nipkow
parents: 26176
diff changeset
  1216
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1217
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1218
lemma multi_subset_induct [consumes 2, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
  1219
assumes "F \<subseteq># A"
nipkow
parents: 26176
diff changeset
  1220
  and empty: "P {#}"
nipkow
parents: 26176
diff changeset
  1221
  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
nipkow
parents: 26176
diff changeset
  1222
shows "P F"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1223
proof -
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1224
  from `F \<subseteq># A`
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1225
  show ?thesis
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1226
  proof (induct F)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1227
    show "P {#}" by fact
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1228
  next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1229
    fix x F
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1230
    assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1231
    show "P (F + {#x#})"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1232
    proof (rule insert)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1233
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1234
      from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1235
      with P show "P F" .
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1236
    qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1237
  qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1238
qed 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1239
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
  1240
text{* A consequence: Extensionality. *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1241
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1242
lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
26178
nipkow
parents: 26176
diff changeset
  1243
apply (rule iffI)
nipkow
parents: 26176
diff changeset
  1244
 prefer 2
nipkow
parents: 26176
diff changeset
  1245
 apply clarsimp 
nipkow
parents: 26176
diff changeset
  1246
apply (induct A arbitrary: B rule: full_multiset_induct)
nipkow
parents: 26176
diff changeset
  1247
apply (rename_tac C)
nipkow
parents: 26176
diff changeset
  1248
apply (case_tac B rule: multiset_cases)
nipkow
parents: 26176
diff changeset
  1249
 apply (simp add: empty_multiset_count)
nipkow
parents: 26176
diff changeset
  1250
apply simp
nipkow
parents: 26176
diff changeset
  1251
apply (case_tac "x \<in># C")
nipkow
parents: 26176
diff changeset
  1252
 apply (force dest: multi_member_split)
nipkow
parents: 26176
diff changeset
  1253
apply (erule_tac x = x in allE)
nipkow
parents: 26176
diff changeset
  1254
apply simp
nipkow
parents: 26176
diff changeset
  1255
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1256
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1257
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1258
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1259
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1260
subsection {* The fold combinator *}
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1261
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1262
text {*
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1263
  The intended behaviour is
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1264
  @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1265
  if @{text f} is associative-commutative. 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1266
*}
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1267
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1268
text {*
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1269
  The graph of @{text "fold_mset"}, @{text "z"}: the start element,
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1270
  @{text "f"}: folding function, @{text "A"}: the multiset, @{text
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1271
  "y"}: the result.
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1272
*}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1273
inductive 
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1274
  fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1275
  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1276
  and z :: 'b
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1277
where
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1278
  emptyI [intro]:  "fold_msetG f z {#} z"
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1279
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1280
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1281
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1282
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1283
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1284
definition
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1285
  fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1286
  "fold_mset f z A = (THE x. fold_msetG f z A x)"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1287
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1288
lemma Diff1_fold_msetG:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1289
  "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
26178
nipkow
parents: 26176
diff changeset
  1290
apply (frule_tac x = x in fold_msetG.insertI)
nipkow
parents: 26176
diff changeset
  1291
apply auto
nipkow
parents: 26176
diff changeset
  1292
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1293