wenzelm@10249
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(* Title: HOL/Library/Multiset.thy
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wenzelm@10249
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ID: $Id$
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paulson@15072
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Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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wenzelm@10249
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*)
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wenzelm@10249
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wenzelm@14706
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header {* Multisets *}
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wenzelm@10249
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nipkow@15131
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theory Multiset
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haftmann@27487
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imports Plain "~~/src/HOL/List"
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nipkow@15131
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begin
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wenzelm@10249
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wenzelm@10249
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subsection {* The type of multisets *}
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nipkow@25162
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typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
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wenzelm@10249
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proof
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nipkow@11464
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show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
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wenzelm@10249
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qed
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wenzelm@10249
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wenzelm@10249
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lemmas multiset_typedef [simp] =
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wenzelm@10277
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Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
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wenzelm@10277
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and [simp] = Rep_multiset_inject [symmetric]
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wenzelm@10249
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wenzelm@19086
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definition
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wenzelm@21404
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Mempty :: "'a multiset" ("{#}") where
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wenzelm@19086
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"{#} = Abs_multiset (\<lambda>a. 0)"
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wenzelm@10249
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wenzelm@21404
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definition
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nipkow@25507
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single :: "'a => 'a multiset" where
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nipkow@25507
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"single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
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wenzelm@10249
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nipkow@26016
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declare
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nipkow@26016
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Mempty_def[code func del] single_def[code func del]
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nipkow@26016
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wenzelm@21404
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definition
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wenzelm@21404
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count :: "'a multiset => 'a => nat" where
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wenzelm@19086
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"count = Rep_multiset"
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wenzelm@10249
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wenzelm@21404
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definition
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wenzelm@21404
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MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
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wenzelm@19086
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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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wenzelm@19086
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wenzelm@19363
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abbreviation
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wenzelm@21404
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Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where
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kleing@25610
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"a :# M == 0 < count M a"
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kleing@25610
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wenzelm@26145
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notation (xsymbols)
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wenzelm@26145
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Melem (infix "\<in>#" 50)
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wenzelm@10249
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wenzelm@10249
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syntax
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nipkow@26033
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"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ :# _./ _#})")
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wenzelm@10249
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translations
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nipkow@26033
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"{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
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wenzelm@19086
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definition
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wenzelm@21404
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set_of :: "'a multiset => 'a set" where
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wenzelm@19086
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"set_of M = {x. x :# M}"
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wenzelm@10249
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haftmann@25571
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instantiation multiset :: (type) "{plus, minus, zero, size}"
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haftmann@25571
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begin
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haftmann@25571
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haftmann@25571
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definition
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nipkow@26016
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union_def[code func del]:
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wenzelm@26145
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"M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
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haftmann@25571
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haftmann@25571
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definition
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wenzelm@26145
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diff_def: "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
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haftmann@25571
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haftmann@25571
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definition
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wenzelm@26145
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Zero_multiset_def [simp]: "0 = {#}"
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haftmann@25571
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haftmann@25571
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definition
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wenzelm@26145
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size_def[code func del]: "size M = setsum (count M) (set_of M)"
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haftmann@25571
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haftmann@25571
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instance ..
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haftmann@25571
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haftmann@25571
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end
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wenzelm@19086
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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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wenzelm@19086
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"multiset_inter A B = A - (A - B)"
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kleing@15869
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wenzelm@26145
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text {* Multiset Enumeration *}
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wenzelm@26145
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syntax
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wenzelm@26176
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"_multiset" :: "args => 'a multiset" ("{#(_)#}")
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nipkow@25507
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translations
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nipkow@25507
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"{#x, xs#}" == "{#x#} + {#xs#}"
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nipkow@25507
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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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nipkow@26178
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by (simp add: multiset_def)
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wenzelm@10249
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nipkow@26016
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lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
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nipkow@26178
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by (simp add: multiset_def)
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nipkow@26016
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lemma union_preserves_multiset:
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nipkow@26178
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"M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
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nipkow@26178
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apply (simp add: multiset_def)
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nipkow@26178
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apply (drule (1) finite_UnI)
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nipkow@26178
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apply (simp del: finite_Un add: Un_def)
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nipkow@26178
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done
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wenzelm@10249
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nipkow@26016
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lemma diff_preserves_multiset:
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nipkow@26178
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"M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
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nipkow@26178
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apply (simp add: multiset_def)
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nipkow@26178
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apply (rule finite_subset)
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nipkow@26178
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apply auto
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nipkow@26178
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done
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wenzelm@10249
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nipkow@26016
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lemma MCollect_preserves_multiset:
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nipkow@26178
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"M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
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nipkow@26178
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apply (simp add: multiset_def)
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nipkow@26178
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apply (rule finite_subset, auto)
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nipkow@26178
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done
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wenzelm@10249
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nipkow@26016
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lemmas in_multiset = const0_in_multiset only1_in_multiset
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nipkow@26016
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union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
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nipkow@26016
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wenzelm@26145
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nipkow@26016
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subsection {* Algebraic properties *}
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wenzelm@10249
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wenzelm@10249
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subsubsection {* Union *}
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wenzelm@10249
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wenzelm@17161
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lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
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nipkow@26178
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by (simp add: union_def Mempty_def in_multiset)
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wenzelm@10249
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wenzelm@17161
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lemma union_commute: "M + N = N + (M::'a multiset)"
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nipkow@26178
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by (simp add: union_def add_ac in_multiset)
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wenzelm@17161
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wenzelm@17161
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lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
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nipkow@26178
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by (simp add: union_def add_ac in_multiset)
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wenzelm@10249
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wenzelm@17161
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lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
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wenzelm@17161
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proof -
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nipkow@26178
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have "M + (N + K) = (N + K) + M" by (rule union_commute)
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nipkow@26178
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also have "\<dots> = N + (K + M)" by (rule union_assoc)
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nipkow@26178
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also have "K + M = M + K" by (rule union_commute)
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wenzelm@17161
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finally show ?thesis .
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wenzelm@17161
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qed
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wenzelm@10249
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wenzelm@17161
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lemmas union_ac = union_assoc union_commute union_lcomm
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wenzelm@10249
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obua@14738
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instance multiset :: (type) comm_monoid_add
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wenzelm@17200
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proof
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obua@14722
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fix a b c :: "'a multiset"
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obua@14722
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show "(a + b) + c = a + (b + c)" by (rule union_assoc)
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obua@14722
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show "a + b = b + a" by (rule union_commute)
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obua@14722
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show "0 + a = a" by simp
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obua@14722
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qed
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wenzelm@10277
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wenzelm@10249
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wenzelm@10249
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subsubsection {* Difference *}
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wenzelm@10249
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wenzelm@17161
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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nipkow@26178
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by (simp add: Mempty_def diff_def in_multiset)
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wenzelm@10249
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wenzelm@17161
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lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
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nipkow@26178
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by (simp add: union_def diff_def in_multiset)
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wenzelm@10249
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bulwahn@26143
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lemma diff_cancel: "A - A = {#}"
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nipkow@26178
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by (simp add: diff_def Mempty_def)
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bulwahn@26143
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wenzelm@10249
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wenzelm@10249
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subsubsection {* Count of elements *}
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wenzelm@10249
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wenzelm@17161
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lemma count_empty [simp]: "count {#} a = 0"
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nipkow@26178
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by (simp add: count_def Mempty_def in_multiset)
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wenzelm@10249
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wenzelm@17161
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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nipkow@26178
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by (simp add: count_def single_def in_multiset)
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wenzelm@10249
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wenzelm@17161
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lemma count_union [simp]: "count (M + N) a = count M a + count N a"
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nipkow@26178
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by (simp add: count_def union_def in_multiset)
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wenzelm@10249
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wenzelm@17161
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
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nipkow@26178
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by (simp add: count_def diff_def in_multiset)
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nipkow@26016
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nipkow@26016
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lemma count_MCollect [simp]:
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nipkow@26178
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"count {# x:#M. P x #} a = (if P a then count M a else 0)"
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nipkow@26178
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by (simp add: count_def MCollect_def in_multiset)
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wenzelm@10249
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wenzelm@10249
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wenzelm@10249
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subsubsection {* Set of elements *}
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wenzelm@10249
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wenzelm@17161
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lemma set_of_empty [simp]: "set_of {#} = {}"
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nipkow@26178
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by (simp add: set_of_def)
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wenzelm@10249
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wenzelm@17161
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lemma set_of_single [simp]: "set_of {#b#} = {b}"
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nipkow@26178
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by (simp add: set_of_def)
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wenzelm@10249
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wenzelm@17161
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lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
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nipkow@26178
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by (auto simp add: set_of_def)
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wenzelm@10249
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wenzelm@17161
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lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
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berghofe@26818
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by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
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wenzelm@10249
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wenzelm@17161
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lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
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nipkow@26178
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by (auto simp add: set_of_def)
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nipkow@26016
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nipkow@26033
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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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nipkow@26178
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by (auto simp add: set_of_def)
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wenzelm@10249
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wenzelm@10249
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wenzelm@10249
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subsubsection {* Size *}
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wenzelm@10249
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nipkow@26016
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lemma size_empty [simp,code func]: "size {#} = 0"
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nipkow@26178
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by (simp add: size_def)
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wenzelm@10249
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nipkow@26016
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lemma size_single [simp,code func]: "size {#b#} = 1"
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nipkow@26178
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by (simp add: size_def)
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wenzelm@10249
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wenzelm@17161
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lemma finite_set_of [iff]: "finite (set_of M)"
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nipkow@26178
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using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
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wenzelm@10249
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wenzelm@17161
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lemma setsum_count_Int:
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nipkow@26178
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"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
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nipkow@26178
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apply (induct rule: finite_induct)
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nipkow@26178
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apply simp
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nipkow@26178
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apply (simp add: Int_insert_left set_of_def)
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nipkow@26178
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done
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wenzelm@10249
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nipkow@26016
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lemma size_union[simp,code func]: "size (M + N::'a multiset) = size M + size N"
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nipkow@26178
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apply (unfold size_def)
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nipkow@26178
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apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
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nipkow@26178
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prefer 2
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nipkow@26178
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apply (rule ext, simp)
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nipkow@26178
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apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
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nipkow@26178
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apply (subst Int_commute)
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nipkow@26178
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apply (simp (no_asm_simp) add: setsum_count_Int)
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nipkow@26178
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done
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wenzelm@10249
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wenzelm@17161
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lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
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nipkow@26178
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apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
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nipkow@26178
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apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
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nipkow@26178
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done
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nipkow@26016
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nipkow@26016
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lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
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nipkow@26178
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by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
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wenzelm@10249
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wenzelm@17161
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lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
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nipkow@26178
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apply (unfold size_def)
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nipkow@26178
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apply (drule setsum_SucD)
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nipkow@26178
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apply auto
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nipkow@26178
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done
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wenzelm@10249
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wenzelm@26145
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wenzelm@10249
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subsubsection {* Equality of multisets *}
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wenzelm@10249
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wenzelm@17161
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lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
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nipkow@26178
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by (simp add: count_def expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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nipkow@26178
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by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
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nipkow@26178
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by (auto simp add: single_def in_multiset expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
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nipkow@26178
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by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
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nipkow@26178
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by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
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nipkow@26178
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by (simp add: union_def in_multiset expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
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nipkow@26178
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by (simp add: union_def in_multiset expand_fun_eq)
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wenzelm@10249
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wenzelm@17161
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lemma union_is_single:
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nipkow@26178
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"(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
|
nipkow@26178
|
276 |
apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
|
nipkow@26178
|
277 |
apply blast
|
nipkow@26178
|
278 |
done
|
wenzelm@10249
|
279 |
|
wenzelm@17161
|
280 |
lemma single_is_union:
|
nipkow@26178
|
281 |
"({#a#} = M + N) \<longleftrightarrow> ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
|
nipkow@26178
|
282 |
apply (unfold Mempty_def single_def union_def)
|
nipkow@26178
|
283 |
apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq)
|
nipkow@26178
|
284 |
apply (blast dest: sym)
|
nipkow@26178
|
285 |
done
|
wenzelm@10249
|
286 |
|
wenzelm@17161
|
287 |
lemma add_eq_conv_diff:
|
wenzelm@10249
|
288 |
"(M + {#a#} = N + {#b#}) =
|
paulson@15072
|
289 |
(M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
|
nipkow@26178
|
290 |
using [[simproc del: neq]]
|
nipkow@26178
|
291 |
apply (unfold single_def union_def diff_def)
|
nipkow@26178
|
292 |
apply (simp (no_asm) add: in_multiset expand_fun_eq)
|
nipkow@26178
|
293 |
apply (rule conjI, force, safe, simp_all)
|
nipkow@26178
|
294 |
apply (simp add: eq_sym_conv)
|
nipkow@26178
|
295 |
done
|
wenzelm@10249
|
296 |
|
kleing@15869
|
297 |
declare Rep_multiset_inject [symmetric, simp del]
|
kleing@15869
|
298 |
|
nipkow@23611
|
299 |
instance multiset :: (type) cancel_ab_semigroup_add
|
nipkow@23611
|
300 |
proof
|
nipkow@23611
|
301 |
fix a b c :: "'a multiset"
|
nipkow@23611
|
302 |
show "a + b = a + c \<Longrightarrow> b = c" by simp
|
nipkow@23611
|
303 |
qed
|
kleing@15869
|
304 |
|
kleing@25610
|
305 |
lemma insert_DiffM:
|
kleing@25610
|
306 |
"x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
|
nipkow@26178
|
307 |
by (clarsimp simp: multiset_eq_conv_count_eq)
|
kleing@25610
|
308 |
|
kleing@25610
|
309 |
lemma insert_DiffM2[simp]:
|
kleing@25610
|
310 |
"x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
|
nipkow@26178
|
311 |
by (clarsimp simp: multiset_eq_conv_count_eq)
|
kleing@25610
|
312 |
|
kleing@25610
|
313 |
lemma multi_union_self_other_eq:
|
kleing@25610
|
314 |
"(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
|
nipkow@26178
|
315 |
by (induct A arbitrary: X Y) auto
|
kleing@25610
|
316 |
|
kleing@25610
|
317 |
lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
|
nipkow@26178
|
318 |
by (metis single_not_empty union_empty union_left_cancel)
|
kleing@25610
|
319 |
|
kleing@25610
|
320 |
lemma insert_noteq_member:
|
kleing@25610
|
321 |
assumes BC: "B + {#b#} = C + {#c#}"
|
kleing@25610
|
322 |
and bnotc: "b \<noteq> c"
|
kleing@25610
|
323 |
shows "c \<in># B"
|
kleing@25610
|
324 |
proof -
|
kleing@25610
|
325 |
have "c \<in># C + {#c#}" by simp
|
kleing@25610
|
326 |
have nc: "\<not> c \<in># {#b#}" using bnotc by simp
|
wenzelm@26145
|
327 |
then have "c \<in># B + {#b#}" using BC by simp
|
wenzelm@26145
|
328 |
then show "c \<in># B" using nc by simp
|
kleing@25610
|
329 |
qed
|
kleing@25610
|
330 |
|
kleing@25610
|
331 |
|
nipkow@26016
|
332 |
lemma add_eq_conv_ex:
|
nipkow@26016
|
333 |
"(M + {#a#} = N + {#b#}) =
|
nipkow@26016
|
334 |
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
|
nipkow@26178
|
335 |
by (auto simp add: add_eq_conv_diff)
|
nipkow@26016
|
336 |
|
nipkow@26016
|
337 |
|
nipkow@26016
|
338 |
lemma empty_multiset_count:
|
nipkow@26016
|
339 |
"(\<forall>x. count A x = 0) = (A = {#})"
|
nipkow@26178
|
340 |
by (metis count_empty multiset_eq_conv_count_eq)
|
nipkow@26016
|
341 |
|
nipkow@26016
|
342 |
|
kleing@15869
|
343 |
subsubsection {* Intersection *}
|
kleing@15869
|
344 |
|
kleing@15869
|
345 |
lemma multiset_inter_count:
|
nipkow@26178
|
346 |
"count (A #\<inter> B) x = min (count A x) (count B x)"
|
nipkow@26178
|
347 |
by (simp add: multiset_inter_def min_def)
|
kleing@15869
|
348 |
|
kleing@15869
|
349 |
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
|
nipkow@26178
|
350 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
|
haftmann@21214
|
351 |
min_max.inf_commute)
|
kleing@15869
|
352 |
|
kleing@15869
|
353 |
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
|
nipkow@26178
|
354 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
|
haftmann@21214
|
355 |
min_max.inf_assoc)
|
kleing@15869
|
356 |
|
kleing@15869
|
357 |
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
|
nipkow@26178
|
358 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
|
kleing@15869
|
359 |
|
wenzelm@17161
|
360 |
lemmas multiset_inter_ac =
|
wenzelm@17161
|
361 |
multiset_inter_commute
|
wenzelm@17161
|
362 |
multiset_inter_assoc
|
wenzelm@17161
|
363 |
multiset_inter_left_commute
|
kleing@15869
|
364 |
|
bulwahn@26143
|
365 |
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
|
nipkow@26178
|
366 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
|
bulwahn@26143
|
367 |
|
kleing@15869
|
368 |
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
|
nipkow@26178
|
369 |
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
|
wenzelm@17161
|
370 |
split: split_if_asm)
|
nipkow@26178
|
371 |
apply clarsimp
|
nipkow@26178
|
372 |
apply (erule_tac x = a in allE)
|
nipkow@26178
|
373 |
apply auto
|
nipkow@26178
|
374 |
done
|
kleing@15869
|
375 |
|
wenzelm@10249
|
376 |
|
nipkow@26016
|
377 |
subsubsection {* Comprehension (filter) *}
|
nipkow@26016
|
378 |
|
nipkow@26016
|
379 |
lemma MCollect_empty[simp, code func]: "MCollect {#} P = {#}"
|
nipkow@26178
|
380 |
by (simp add: MCollect_def Mempty_def Abs_multiset_inject
|
wenzelm@26145
|
381 |
in_multiset expand_fun_eq)
|
nipkow@26016
|
382 |
|
nipkow@26016
|
383 |
lemma MCollect_single[simp, code func]:
|
nipkow@26178
|
384 |
"MCollect {#x#} P = (if P x then {#x#} else {#})"
|
nipkow@26178
|
385 |
by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
|
wenzelm@26145
|
386 |
in_multiset expand_fun_eq)
|
nipkow@26016
|
387 |
|
nipkow@26016
|
388 |
lemma MCollect_union[simp, code func]:
|
nipkow@26016
|
389 |
"MCollect (M+N) f = MCollect M f + MCollect N f"
|
nipkow@26178
|
390 |
by (simp add: MCollect_def union_def Abs_multiset_inject
|
wenzelm@26145
|
391 |
in_multiset expand_fun_eq)
|
nipkow@26016
|
392 |
|
nipkow@26016
|
393 |
|
nipkow@26016
|
394 |
subsection {* Induction and case splits *}
|
wenzelm@10249
|
395 |
|
wenzelm@10249
|
396 |
lemma setsum_decr:
|
wenzelm@11701
|
397 |
"finite F ==> (0::nat) < f a ==>
|
paulson@15072
|
398 |
setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
|
nipkow@26178
|
399 |
apply (induct rule: finite_induct)
|
nipkow@26178
|
400 |
apply auto
|
nipkow@26178
|
401 |
apply (drule_tac a = a in mk_disjoint_insert, auto)
|
nipkow@26178
|
402 |
done
|
wenzelm@10249
|
403 |
|
wenzelm@10313
|
404 |
lemma rep_multiset_induct_aux:
|
nipkow@26178
|
405 |
assumes 1: "P (\<lambda>a. (0::nat))"
|
nipkow@26178
|
406 |
and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
|
nipkow@26178
|
407 |
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
|
nipkow@26178
|
408 |
apply (unfold multiset_def)
|
nipkow@26178
|
409 |
apply (induct_tac n, simp, clarify)
|
nipkow@26178
|
410 |
apply (subgoal_tac "f = (\<lambda>a.0)")
|
nipkow@26178
|
411 |
apply simp
|
nipkow@26178
|
412 |
apply (rule 1)
|
nipkow@26178
|
413 |
apply (rule ext, force, clarify)
|
nipkow@26178
|
414 |
apply (frule setsum_SucD, clarify)
|
nipkow@26178
|
415 |
apply (rename_tac a)
|
nipkow@26178
|
416 |
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
|
nipkow@26178
|
417 |
prefer 2
|
nipkow@26178
|
418 |
apply (rule finite_subset)
|
nipkow@26178
|
419 |
prefer 2
|
nipkow@26178
|
420 |
apply assumption
|
nipkow@26178
|
421 |
apply simp
|
nipkow@26178
|
422 |
apply blast
|
nipkow@26178
|
423 |
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
|
nipkow@26178
|
424 |
prefer 2
|
nipkow@26178
|
425 |
apply (rule ext)
|
nipkow@26178
|
426 |
apply (simp (no_asm_simp))
|
nipkow@26178
|
427 |
apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
|
nipkow@26178
|
428 |
apply (erule allE, erule impE, erule_tac [2] mp, blast)
|
nipkow@26178
|
429 |
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
|
nipkow@26178
|
430 |
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
|
nipkow@26178
|
431 |
prefer 2
|
nipkow@26178
|
432 |
apply blast
|
nipkow@26178
|
433 |
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
|
nipkow@26178
|
434 |
prefer 2
|
nipkow@26178
|
435 |
apply blast
|
nipkow@26178
|
436 |
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
|
nipkow@26178
|
437 |
done
|
wenzelm@10249
|
438 |
|
wenzelm@10313
|
439 |
theorem rep_multiset_induct:
|
nipkow@11464
|
440 |
"f \<in> multiset ==> P (\<lambda>a. 0) ==>
|
wenzelm@11701
|
441 |
(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
|
nipkow@26178
|
442 |
using rep_multiset_induct_aux by blast
|
wenzelm@10249
|
443 |
|
wenzelm@18258
|
444 |
theorem multiset_induct [case_names empty add, induct type: multiset]:
|
nipkow@26178
|
445 |
assumes empty: "P {#}"
|
nipkow@26178
|
446 |
and add: "!!M x. P M ==> P (M + {#x#})"
|
nipkow@26178
|
447 |
shows "P M"
|
wenzelm@10249
|
448 |
proof -
|
wenzelm@10249
|
449 |
note defns = union_def single_def Mempty_def
|
wenzelm@10249
|
450 |
show ?thesis
|
wenzelm@10249
|
451 |
apply (rule Rep_multiset_inverse [THEN subst])
|
wenzelm@10313
|
452 |
apply (rule Rep_multiset [THEN rep_multiset_induct])
|
wenzelm@18258
|
453 |
apply (rule empty [unfolded defns])
|
paulson@15072
|
454 |
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
|
wenzelm@10249
|
455 |
prefer 2
|
wenzelm@10249
|
456 |
apply (simp add: expand_fun_eq)
|
wenzelm@10249
|
457 |
apply (erule ssubst)
|
wenzelm@17200
|
458 |
apply (erule Abs_multiset_inverse [THEN subst])
|
nipkow@26016
|
459 |
apply (drule add [unfolded defns, simplified])
|
nipkow@26016
|
460 |
apply(simp add:in_multiset)
|
wenzelm@10249
|
461 |
done
|
wenzelm@10249
|
462 |
qed
|
wenzelm@10249
|
463 |
|
kleing@25610
|
464 |
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
|
nipkow@26178
|
465 |
by (induct M) auto
|
kleing@25610
|
466 |
|
kleing@25610
|
467 |
lemma multiset_cases [cases type, case_names empty add]:
|
nipkow@26178
|
468 |
assumes em: "M = {#} \<Longrightarrow> P"
|
nipkow@26178
|
469 |
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
|
nipkow@26178
|
470 |
shows "P"
|
kleing@25610
|
471 |
proof (cases "M = {#}")
|
wenzelm@26145
|
472 |
assume "M = {#}" then show ?thesis using em by simp
|
kleing@25610
|
473 |
next
|
kleing@25610
|
474 |
assume "M \<noteq> {#}"
|
kleing@25610
|
475 |
then obtain M' m where "M = M' + {#m#}"
|
kleing@25610
|
476 |
by (blast dest: multi_nonempty_split)
|
wenzelm@26145
|
477 |
then show ?thesis using add by simp
|
kleing@25610
|
478 |
qed
|
kleing@25610
|
479 |
|
kleing@25610
|
480 |
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
|
nipkow@26178
|
481 |
apply (cases M)
|
nipkow@26178
|
482 |
apply simp
|
nipkow@26178
|
483 |
apply (rule_tac x="M - {#x#}" in exI, simp)
|
nipkow@26178
|
484 |
done
|
kleing@25610
|
485 |
|
nipkow@26033
|
486 |
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
|
nipkow@26178
|
487 |
apply (subst multiset_eq_conv_count_eq)
|
nipkow@26178
|
488 |
apply auto
|
nipkow@26178
|
489 |
done
|
wenzelm@10249
|
490 |
|
kleing@15869
|
491 |
declare multiset_typedef [simp del]
|
wenzelm@10249
|
492 |
|
kleing@25610
|
493 |
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
|
nipkow@26178
|
494 |
by (cases "B = {#}") (auto dest: multi_member_split)
|
wenzelm@26145
|
495 |
|
wenzelm@17161
|
496 |
|
nipkow@26016
|
497 |
subsection {* Orderings *}
|
wenzelm@10249
|
498 |
|
wenzelm@10249
|
499 |
subsubsection {* Well-foundedness *}
|
wenzelm@10249
|
500 |
|
wenzelm@19086
|
501 |
definition
|
berghofe@23751
|
502 |
mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
|
haftmann@27106
|
503 |
[code func del]:"mult1 r =
|
berghofe@23751
|
504 |
{(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
|
berghofe@23751
|
505 |
(\<forall>b. b :# K --> (b, a) \<in> r)}"
|
wenzelm@10249
|
506 |
|
wenzelm@21404
|
507 |
definition
|
berghofe@23751
|
508 |
mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
|
berghofe@23751
|
509 |
"mult r = (mult1 r)\<^sup>+"
|
wenzelm@10249
|
510 |
|
berghofe@23751
|
511 |
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
|
nipkow@26178
|
512 |
by (simp add: mult1_def)
|
wenzelm@10249
|
513 |
|
berghofe@23751
|
514 |
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
|
berghofe@23751
|
515 |
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
|
berghofe@23751
|
516 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
|
wenzelm@19582
|
517 |
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
|
wenzelm@10249
|
518 |
proof (unfold mult1_def)
|
berghofe@23751
|
519 |
let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
|
nipkow@11464
|
520 |
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
|
berghofe@23751
|
521 |
let ?case1 = "?case1 {(N, M). ?R N M}"
|
wenzelm@10249
|
522 |
|
berghofe@23751
|
523 |
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
|
wenzelm@18258
|
524 |
then have "\<exists>a' M0' K.
|
nipkow@11464
|
525 |
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
|
wenzelm@18258
|
526 |
then show "?case1 \<or> ?case2"
|
wenzelm@10249
|
527 |
proof (elim exE conjE)
|
wenzelm@10249
|
528 |
fix a' M0' K
|
wenzelm@10249
|
529 |
assume N: "N = M0' + K" and r: "?r K a'"
|
wenzelm@10249
|
530 |
assume "M0 + {#a#} = M0' + {#a'#}"
|
wenzelm@18258
|
531 |
then have "M0 = M0' \<and> a = a' \<or>
|
nipkow@11464
|
532 |
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
|
wenzelm@10249
|
533 |
by (simp only: add_eq_conv_ex)
|
wenzelm@18258
|
534 |
then show ?thesis
|
wenzelm@10249
|
535 |
proof (elim disjE conjE exE)
|
wenzelm@10249
|
536 |
assume "M0 = M0'" "a = a'"
|
nipkow@11464
|
537 |
with N r have "?r K a \<and> N = M0 + K" by simp
|
wenzelm@18258
|
538 |
then have ?case2 .. then show ?thesis ..
|
wenzelm@10249
|
539 |
next
|
wenzelm@10249
|
540 |
fix K'
|
wenzelm@10249
|
541 |
assume "M0' = K' + {#a#}"
|
wenzelm@10249
|
542 |
with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
|
wenzelm@10249
|
543 |
|
wenzelm@10249
|
544 |
assume "M0 = K' + {#a'#}"
|
wenzelm@10249
|
545 |
with r have "?R (K' + K) M0" by blast
|
wenzelm@18258
|
546 |
with n have ?case1 by simp then show ?thesis ..
|
wenzelm@10249
|
547 |
qed
|
wenzelm@10249
|
548 |
qed
|
wenzelm@10249
|
549 |
qed
|
wenzelm@10249
|
550 |
|
berghofe@23751
|
551 |
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
|
wenzelm@10249
|
552 |
proof
|
wenzelm@10249
|
553 |
let ?R = "mult1 r"
|
wenzelm@10249
|
554 |
let ?W = "acc ?R"
|
wenzelm@10249
|
555 |
{
|
wenzelm@10249
|
556 |
fix M M0 a
|
berghofe@23751
|
557 |
assume M0: "M0 \<in> ?W"
|
berghofe@23751
|
558 |
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
|
berghofe@23751
|
559 |
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
|
berghofe@23751
|
560 |
have "M0 + {#a#} \<in> ?W"
|
berghofe@23751
|
561 |
proof (rule accI [of "M0 + {#a#}"])
|
wenzelm@10249
|
562 |
fix N
|
berghofe@23751
|
563 |
assume "(N, M0 + {#a#}) \<in> ?R"
|
berghofe@23751
|
564 |
then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
|
berghofe@23751
|
565 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
|
wenzelm@10249
|
566 |
by (rule less_add)
|
berghofe@23751
|
567 |
then show "N \<in> ?W"
|
wenzelm@10249
|
568 |
proof (elim exE disjE conjE)
|
berghofe@23751
|
569 |
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
|
berghofe@23751
|
570 |
from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
|
berghofe@23751
|
571 |
from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
|
berghofe@23751
|
572 |
then show "N \<in> ?W" by (simp only: N)
|
wenzelm@10249
|
573 |
next
|
wenzelm@10249
|
574 |
fix K
|
wenzelm@10249
|
575 |
assume N: "N = M0 + K"
|
berghofe@23751
|
576 |
assume "\<forall>b. b :# K --> (b, a) \<in> r"
|
berghofe@23751
|
577 |
then have "M0 + K \<in> ?W"
|
wenzelm@10249
|
578 |
proof (induct K)
|
wenzelm@18730
|
579 |
case empty
|
berghofe@23751
|
580 |
from M0 show "M0 + {#} \<in> ?W" by simp
|
wenzelm@18730
|
581 |
next
|
wenzelm@18730
|
582 |
case (add K x)
|
berghofe@23751
|
583 |
from add.prems have "(x, a) \<in> r" by simp
|
berghofe@23751
|
584 |
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
|
berghofe@23751
|
585 |
moreover from add have "M0 + K \<in> ?W" by simp
|
berghofe@23751
|
586 |
ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
|
berghofe@23751
|
587 |
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
|
wenzelm@10249
|
588 |
qed
|
berghofe@23751
|
589 |
then show "N \<in> ?W" by (simp only: N)
|
wenzelm@10249
|
590 |
qed
|
wenzelm@10249
|
591 |
qed
|
wenzelm@10249
|
592 |
} note tedious_reasoning = this
|
wenzelm@10249
|
593 |
|
berghofe@23751
|
594 |
assume wf: "wf r"
|
wenzelm@10249
|
595 |
fix M
|
berghofe@23751
|
596 |
show "M \<in> ?W"
|
wenzelm@10249
|
597 |
proof (induct M)
|
berghofe@23751
|
598 |
show "{#} \<in> ?W"
|
wenzelm@10249
|
599 |
proof (rule accI)
|
berghofe@23751
|
600 |
fix b assume "(b, {#}) \<in> ?R"
|
berghofe@23751
|
601 |
with not_less_empty show "b \<in> ?W" by contradiction
|
wenzelm@10249
|
602 |
qed
|
wenzelm@10249
|
603 |
|
berghofe@23751
|
604 |
fix M a assume "M \<in> ?W"
|
berghofe@23751
|
605 |
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
|
wenzelm@10249
|
606 |
proof induct
|
wenzelm@10249
|
607 |
fix a
|
berghofe@23751
|
608 |
assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
|
berghofe@23751
|
609 |
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
|
wenzelm@10249
|
610 |
proof
|
berghofe@23751
|
611 |
fix M assume "M \<in> ?W"
|
berghofe@23751
|
612 |
then show "M + {#a#} \<in> ?W"
|
wenzelm@23373
|
613 |
by (rule acc_induct) (rule tedious_reasoning [OF _ r])
|
wenzelm@10249
|
614 |
qed
|
wenzelm@10249
|
615 |
qed
|
berghofe@23751
|
616 |
from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
|
wenzelm@10249
|
617 |
qed
|
wenzelm@10249
|
618 |
qed
|
wenzelm@10249
|
619 |
|
berghofe@23751
|
620 |
theorem wf_mult1: "wf r ==> wf (mult1 r)"
|
nipkow@26178
|
621 |
by (rule acc_wfI) (rule all_accessible)
|
wenzelm@10249
|
622 |
|
berghofe@23751
|
623 |
theorem wf_mult: "wf r ==> wf (mult r)"
|
nipkow@26178
|
624 |
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
|
wenzelm@10249
|
625 |
|
wenzelm@10249
|
626 |
|
wenzelm@10249
|
627 |
subsubsection {* Closure-free presentation *}
|
wenzelm@10249
|
628 |
|
wenzelm@10249
|
629 |
(*Badly needed: a linear arithmetic procedure for multisets*)
|
wenzelm@10249
|
630 |
|
wenzelm@10249
|
631 |
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
|
nipkow@26178
|
632 |
by (simp add: multiset_eq_conv_count_eq)
|
wenzelm@10249
|
633 |
|
wenzelm@10249
|
634 |
text {* One direction. *}
|
wenzelm@10249
|
635 |
|
wenzelm@10249
|
636 |
lemma mult_implies_one_step:
|
berghofe@23751
|
637 |
"trans r ==> (M, N) \<in> mult r ==>
|
nipkow@11464
|
638 |
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
|
berghofe@23751
|
639 |
(\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
|
nipkow@26178
|
640 |
apply (unfold mult_def mult1_def set_of_def)
|
nipkow@26178
|
641 |
apply (erule converse_trancl_induct, clarify)
|
nipkow@26178
|
642 |
apply (rule_tac x = M0 in exI, simp, clarify)
|
nipkow@26178
|
643 |
apply (case_tac "a :# K")
|
nipkow@26178
|
644 |
apply (rule_tac x = I in exI)
|
nipkow@26178
|
645 |
apply (simp (no_asm))
|
nipkow@26178
|
646 |
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
|
nipkow@26178
|
647 |
apply (simp (no_asm_simp) add: union_assoc [symmetric])
|
nipkow@26178
|
648 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
|
nipkow@26178
|
649 |
apply (simp add: diff_union_single_conv)
|
nipkow@26178
|
650 |
apply (simp (no_asm_use) add: trans_def)
|
nipkow@26178
|
651 |
apply blast
|
nipkow@26178
|
652 |
apply (subgoal_tac "a :# I")
|
nipkow@26178
|
653 |
apply (rule_tac x = "I - {#a#}" in exI)
|
nipkow@26178
|
654 |
apply (rule_tac x = "J + {#a#}" in exI)
|
nipkow@26178
|
655 |
apply (rule_tac x = "K + Ka" in exI)
|
nipkow@26178
|
656 |
apply (rule conjI)
|
nipkow@26178
|
657 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
|
nipkow@26178
|
658 |
apply (rule conjI)
|
nipkow@26178
|
659 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
|
nipkow@26178
|
660 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
|
nipkow@26178
|
661 |
apply (simp (no_asm_use) add: trans_def)
|
nipkow@26178
|
662 |
apply blast
|
nipkow@26178
|
663 |
apply (subgoal_tac "a :# (M0 + {#a#})")
|
nipkow@26178
|
664 |
apply simp
|
nipkow@26178
|
665 |
apply (simp (no_asm))
|
nipkow@26178
|
666 |
done
|
wenzelm@10249
|
667 |
|
wenzelm@10249
|
668 |
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
|
nipkow@26178
|
669 |
by (simp add: multiset_eq_conv_count_eq)
|
wenzelm@10249
|
670 |
|
nipkow@11464
|
671 |
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
|
nipkow@26178
|
672 |
apply (erule size_eq_Suc_imp_elem [THEN exE])
|
nipkow@26178
|
673 |
apply (drule elem_imp_eq_diff_union, auto)
|
nipkow@26178
|
674 |
done
|
wenzelm@10249
|
675 |
|
wenzelm@10249
|
676 |
lemma one_step_implies_mult_aux:
|
berghofe@23751
|
677 |
"trans r ==>
|
berghofe@23751
|
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))
|
berghofe@23751
|
679 |
--> (I + K, I + J) \<in> mult r"
|
nipkow@26178
|
680 |
apply (induct_tac n, auto)
|
nipkow@26178
|
681 |
apply (frule size_eq_Suc_imp_eq_union, clarify)
|
nipkow@26178
|
682 |
apply (rename_tac "J'", simp)
|
nipkow@26178
|
683 |
apply (erule notE, auto)
|
nipkow@26178
|
684 |
apply (case_tac "J' = {#}")
|
nipkow@26178
|
685 |
apply (simp add: mult_def)
|
nipkow@26178
|
686 |
apply (rule r_into_trancl)
|
nipkow@26178
|
687 |
apply (simp add: mult1_def set_of_def, blast)
|
nipkow@26178
|
688 |
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
|
nipkow@26178
|
689 |
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
|
nipkow@26178
|
690 |
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
|
nipkow@26178
|
691 |
apply (erule ssubst)
|
nipkow@26178
|
692 |
apply (simp add: Ball_def, auto)
|
nipkow@26178
|
693 |
apply (subgoal_tac
|
nipkow@26178
|
694 |
"((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
|
nipkow@26178
|
695 |
(I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
|
nipkow@26178
|
696 |
prefer 2
|
nipkow@26178
|
697 |
apply force
|
nipkow@26178
|
698 |
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
|
nipkow@26178
|
699 |
apply (erule trancl_trans)
|
nipkow@26178
|
700 |
apply (rule r_into_trancl)
|
nipkow@26178
|
701 |
apply (simp add: mult1_def set_of_def)
|
nipkow@26178
|
702 |
apply (rule_tac x = a in exI)
|
nipkow@26178
|
703 |
apply (rule_tac x = "I + J'" in exI)
|
nipkow@26178
|
704 |
apply (simp add: union_ac)
|
nipkow@26178
|
705 |
done
|
wenzelm@10249
|
706 |
|
wenzelm@17161
|
707 |
lemma one_step_implies_mult:
|
berghofe@23751
|
708 |
"trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
|
berghofe@23751
|
709 |
==> (I + K, I + J) \<in> mult r"
|
nipkow@26178
|
710 |
using one_step_implies_mult_aux by blast
|
wenzelm@10249
|
711 |
|
wenzelm@10249
|
712 |
|
wenzelm@10249
|
713 |
subsubsection {* Partial-order properties *}
|
wenzelm@10249
|
714 |
|
haftmann@26567
|
715 |
instantiation multiset :: (order) order
|
haftmann@26567
|
716 |
begin
|
wenzelm@10249
|
717 |
|
haftmann@26567
|
718 |
definition
|
haftmann@27106
|
719 |
less_multiset_def [code func del]: "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
|
haftmann@26567
|
720 |
|
haftmann@26567
|
721 |
definition
|
haftmann@27106
|
722 |
le_multiset_def [code func del]: "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
|
wenzelm@10249
|
723 |
|
berghofe@23751
|
724 |
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
|
nipkow@26178
|
725 |
unfolding trans_def by (blast intro: order_less_trans)
|
wenzelm@10249
|
726 |
|
wenzelm@10249
|
727 |
text {*
|
wenzelm@10249
|
728 |
\medskip Irreflexivity.
|
wenzelm@10249
|
729 |
*}
|
wenzelm@10249
|
730 |
|
wenzelm@10249
|
731 |
lemma mult_irrefl_aux:
|
nipkow@26178
|
732 |
"finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
|
nipkow@26178
|
733 |
by (induct rule: finite_induct) (auto intro: order_less_trans)
|
wenzelm@10249
|
734 |
|
wenzelm@17161
|
735 |
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
|
nipkow@26178
|
736 |
apply (unfold less_multiset_def, auto)
|
nipkow@26178
|
737 |
apply (drule trans_base_order [THEN mult_implies_one_step], auto)
|
nipkow@26178
|
738 |
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
|
nipkow@26178
|
739 |
apply (simp add: set_of_eq_empty_iff)
|
nipkow@26178
|
740 |
done
|
wenzelm@10249
|
741 |
|
wenzelm@10249
|
742 |
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
|
nipkow@26178
|
743 |
using insert mult_less_not_refl by fast
|
wenzelm@10249
|
744 |
|
wenzelm@10249
|
745 |
|
wenzelm@10249
|
746 |
text {* Transitivity. *}
|
wenzelm@10249
|
747 |
|
wenzelm@10249
|
748 |
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
|
nipkow@26178
|
749 |
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
|
wenzelm@10249
|
750 |
|
wenzelm@10249
|
751 |
text {* Asymmetry. *}
|
wenzelm@10249
|
752 |
|
nipkow@11464
|
753 |
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
|
nipkow@26178
|
754 |
apply auto
|
nipkow@26178
|
755 |
apply (rule mult_less_not_refl [THEN notE])
|
nipkow@26178
|
756 |
apply (erule mult_less_trans, assumption)
|
nipkow@26178
|
757 |
done
|
wenzelm@10249
|
758 |
|
wenzelm@10249
|
759 |
theorem mult_less_asym:
|
nipkow@26178
|
760 |
"M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
|
nipkow@26178
|
761 |
using mult_less_not_sym by blast
|
wenzelm@10249
|
762 |
|
wenzelm@10249
|
763 |
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
|
nipkow@26178
|
764 |
unfolding le_multiset_def by auto
|
wenzelm@10249
|
765 |
|
wenzelm@10249
|
766 |
text {* Anti-symmetry. *}
|
wenzelm@10249
|
767 |
|
wenzelm@10249
|
768 |
theorem mult_le_antisym:
|
nipkow@26178
|
769 |
"M <= N ==> N <= M ==> M = (N::'a::order multiset)"
|
nipkow@26178
|
770 |
unfolding le_multiset_def by (blast dest: mult_less_not_sym)
|
wenzelm@10249
|
771 |
|
wenzelm@10249
|
772 |
text {* Transitivity. *}
|
wenzelm@10249
|
773 |
|
wenzelm@10249
|
774 |
theorem mult_le_trans:
|
nipkow@26178
|
775 |
"K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
|
nipkow@26178
|
776 |
unfolding le_multiset_def by (blast intro: mult_less_trans)
|
wenzelm@10249
|
777 |
|
wenzelm@11655
|
778 |
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
|
nipkow@26178
|
779 |
unfolding le_multiset_def by auto
|
wenzelm@10249
|
780 |
|
haftmann@26567
|
781 |
instance
|
nipkow@26178
|
782 |
apply intro_classes
|
nipkow@26178
|
783 |
apply (rule mult_less_le)
|
nipkow@26178
|
784 |
apply (rule mult_le_refl)
|
nipkow@26178
|
785 |
apply (erule mult_le_trans, assumption)
|
nipkow@26178
|
786 |
apply (erule mult_le_antisym, assumption)
|
nipkow@26178
|
787 |
done
|
wenzelm@10277
|
788 |
|
haftmann@26567
|
789 |
end
|
haftmann@26567
|
790 |
|
wenzelm@10249
|
791 |
|
wenzelm@10249
|
792 |
subsubsection {* Monotonicity of multiset union *}
|
wenzelm@10249
|
793 |
|
wenzelm@17161
|
794 |
lemma mult1_union:
|
nipkow@26178
|
795 |
"(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
|
nipkow@26178
|
796 |
apply (unfold mult1_def)
|
nipkow@26178
|
797 |
apply auto
|
nipkow@26178
|
798 |
apply (rule_tac x = a in exI)
|
nipkow@26178
|
799 |
apply (rule_tac x = "C + M0" in exI)
|
nipkow@26178
|
800 |
apply (simp add: union_assoc)
|
nipkow@26178
|
801 |
done
|
wenzelm@10249
|
802 |
|
wenzelm@10249
|
803 |
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
|
nipkow@26178
|
804 |
apply (unfold less_multiset_def mult_def)
|
nipkow@26178
|
805 |
apply (erule trancl_induct)
|
nipkow@26178
|
806 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
|
nipkow@26178
|
807 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
|
nipkow@26178
|
808 |
done
|
wenzelm@10249
|
809 |
|
wenzelm@10249
|
810 |
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
|
nipkow@26178
|
811 |
apply (subst union_commute [of B C])
|
nipkow@26178
|
812 |
apply (subst union_commute [of D C])
|
nipkow@26178
|
813 |
apply (erule union_less_mono2)
|
nipkow@26178
|
814 |
done
|
wenzelm@10249
|
815 |
|
wenzelm@17161
|
816 |
lemma union_less_mono:
|
nipkow@26178
|
817 |
"A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
|
nipkow@26178
|
818 |
by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
|
wenzelm@10249
|
819 |
|
wenzelm@17161
|
820 |
lemma union_le_mono:
|
nipkow@26178
|
821 |
"A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
|
nipkow@26178
|
822 |
unfolding le_multiset_def
|
nipkow@26178
|
823 |
by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
|
wenzelm@10249
|
824 |
|
wenzelm@17161
|
825 |
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
|
nipkow@26178
|
826 |
apply (unfold le_multiset_def less_multiset_def)
|
nipkow@26178
|
827 |
apply (case_tac "M = {#}")
|
nipkow@26178
|
828 |
prefer 2
|
nipkow@26178
|
829 |
apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
|
nipkow@26178
|
830 |
prefer 2
|
nipkow@26178
|
831 |
apply (rule one_step_implies_mult)
|
nipkow@26178
|
832 |
apply (simp only: trans_def)
|
nipkow@26178
|
833 |
apply auto
|
nipkow@26178
|
834 |
done
|
wenzelm@10249
|
835 |
|
wenzelm@17161
|
836 |
lemma union_upper1: "A <= A + (B::'a::order multiset)"
|
paulson@15072
|
837 |
proof -
|
wenzelm@17200
|
838 |
have "A + {#} <= A + B" by (blast intro: union_le_mono)
|
wenzelm@18258
|
839 |
then show ?thesis by simp
|
paulson@15072
|
840 |
qed
|
paulson@15072
|
841 |
|
wenzelm@17161
|
842 |
lemma union_upper2: "B <= A + (B::'a::order multiset)"
|
nipkow@26178
|
843 |
by (subst union_commute) (rule union_upper1)
|
paulson@15072
|
844 |
|
nipkow@23611
|
845 |
instance multiset :: (order) pordered_ab_semigroup_add
|
nipkow@26178
|
846 |
apply intro_classes
|
nipkow@26178
|
847 |
apply (erule union_le_mono[OF mult_le_refl])
|
nipkow@26178
|
848 |
done
|
wenzelm@26145
|
849 |
|
paulson@15072
|
850 |
|
wenzelm@17200
|
851 |
subsection {* Link with lists *}
|
paulson@15072
|
852 |
|
nipkow@26016
|
853 |
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
|
wenzelm@26145
|
854 |
"multiset_of [] = {#}" |
|
wenzelm@26145
|
855 |
"multiset_of (a # x) = multiset_of x + {# a #}"
|
paulson@15072
|
856 |
|
paulson@15072
|
857 |
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
|
nipkow@26178
|
858 |
by (induct x) auto
|
paulson@15072
|
859 |
|
paulson@15072
|
860 |
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
|
nipkow@26178
|
861 |
by (induct x) auto
|
paulson@15072
|
862 |
|
paulson@15072
|
863 |
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
|
nipkow@26178
|
864 |
by (induct x) auto
|
kleing@15867
|
865 |
|
kleing@15867
|
866 |
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
|
nipkow@26178
|
867 |
by (induct xs) auto
|
paulson@15072
|
868 |
|
wenzelm@18258
|
869 |
lemma multiset_of_append [simp]:
|
nipkow@26178
|
870 |
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
|
nipkow@26178
|
871 |
by (induct xs arbitrary: ys) (auto simp: union_ac)
|
wenzelm@18730
|
872 |
|
paulson@15072
|
873 |
lemma surj_multiset_of: "surj multiset_of"
|
nipkow@26178
|
874 |
apply (unfold surj_def)
|
nipkow@26178
|
875 |
apply (rule allI)
|
nipkow@26178
|
876 |
apply (rule_tac M = y in multiset_induct)
|
nipkow@26178
|
877 |
apply auto
|
nipkow@26178
|
878 |
apply (rule_tac x = "x # xa" in exI)
|
nipkow@26178
|
879 |
apply auto
|
nipkow@26178
|
880 |
done
|
wenzelm@10249
|
881 |
|
nipkow@25162
|
882 |
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
|
nipkow@26178
|
883 |
by (induct x) auto
|
paulson@15072
|
884 |
|
wenzelm@17200
|
885 |
lemma distinct_count_atmost_1:
|
nipkow@26178
|
886 |
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
|
nipkow@26178
|
887 |
apply (induct x, simp, rule iffI, simp_all)
|
nipkow@26178
|
888 |
apply (rule conjI)
|
nipkow@26178
|
889 |
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
|
nipkow@26178
|
890 |
apply (erule_tac x = a in allE, simp, clarify)
|
nipkow@26178
|
891 |
apply (erule_tac x = aa in allE, simp)
|
nipkow@26178
|
892 |
done
|
paulson@15072
|
893 |
|
wenzelm@17200
|
894 |
lemma multiset_of_eq_setD:
|
kleing@15867
|
895 |
"multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
|
nipkow@26178
|
896 |
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
|
kleing@15867
|
897 |
|
wenzelm@17200
|
898 |
lemma set_eq_iff_multiset_of_eq_distinct:
|
wenzelm@26145
|
899 |
"distinct x \<Longrightarrow> distinct y \<Longrightarrow>
|
wenzelm@26145
|
900 |
(set x = set y) = (multiset_of x = multiset_of y)"
|
nipkow@26178
|
901 |
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
|
paulson@15072
|
902 |
|
wenzelm@17200
|
903 |
lemma set_eq_iff_multiset_of_remdups_eq:
|
paulson@15072
|
904 |
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
|
nipkow@26178
|
905 |
apply (rule iffI)
|
nipkow@26178
|
906 |
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
|
nipkow@26178
|
907 |
apply (drule distinct_remdups [THEN distinct_remdups
|
wenzelm@26145
|
908 |
[THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
|
nipkow@26178
|
909 |
apply simp
|
nipkow@26178
|
910 |
done
|
wenzelm@10249
|
911 |
|
wenzelm@18258
|
912 |
lemma multiset_of_compl_union [simp]:
|
nipkow@26178
|
913 |
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
|
nipkow@26178
|
914 |
by (induct xs) (auto simp: union_ac)
|
paulson@15072
|
915 |
|
wenzelm@17200
|
916 |
lemma count_filter:
|
nipkow@26178
|
917 |
"count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
|
nipkow@26178
|
918 |
by (induct xs) auto
|
kleing@15867
|
919 |
|
bulwahn@26143
|
920 |
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
|
nipkow@26178
|
921 |
apply (induct ls arbitrary: i)
|
nipkow@26178
|
922 |
apply simp
|
nipkow@26178
|
923 |
apply (case_tac i)
|
nipkow@26178
|
924 |
apply auto
|
nipkow@26178
|
925 |
done
|
bulwahn@26143
|
926 |
|
bulwahn@26143
|
927 |
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
|
nipkow@26178
|
928 |
by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
|
bulwahn@26143
|
929 |
|
bulwahn@26143
|
930 |
lemma multiset_of_eq_length:
|
nipkow@26178
|
931 |
assumes "multiset_of xs = multiset_of ys"
|
nipkow@26178
|
932 |
shows "length xs = length ys"
|
nipkow@26178
|
933 |
using assms
|
bulwahn@26143
|
934 |
proof (induct arbitrary: ys rule: length_induct)
|
bulwahn@26143
|
935 |
case (1 xs ys)
|
bulwahn@26143
|
936 |
show ?case
|
bulwahn@26143
|
937 |
proof (cases xs)
|
wenzelm@26145
|
938 |
case Nil with "1.prems" show ?thesis by simp
|
bulwahn@26143
|
939 |
next
|
bulwahn@26143
|
940 |
case (Cons x xs')
|
bulwahn@26143
|
941 |
note xCons = Cons
|
bulwahn@26143
|
942 |
show ?thesis
|
bulwahn@26143
|
943 |
proof (cases ys)
|
bulwahn@26143
|
944 |
case Nil
|
wenzelm@26145
|
945 |
with "1.prems" Cons show ?thesis by simp
|
bulwahn@26143
|
946 |
next
|
bulwahn@26143
|
947 |
case (Cons y ys')
|
bulwahn@26143
|
948 |
have x_in_ys: "x = y \<or> x \<in> set ys'"
|
bulwahn@26143
|
949 |
proof (cases "x = y")
|
wenzelm@26145
|
950 |
case True then show ?thesis ..
|
bulwahn@26143
|
951 |
next
|
bulwahn@26143
|
952 |
case False
|
wenzelm@26145
|
953 |
from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
|
bulwahn@26143
|
954 |
with False show ?thesis by (simp add: mem_set_multiset_eq)
|
bulwahn@26143
|
955 |
qed
|
wenzelm@26145
|
956 |
from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
|
wenzelm@26145
|
957 |
(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
|
wenzelm@26145
|
958 |
from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
|
bulwahn@26143
|
959 |
apply -
|
bulwahn@26143
|
960 |
apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
|
bulwahn@26143
|
961 |
apply fastsimp
|
bulwahn@26143
|
962 |
done
|
wenzelm@26145
|
963 |
with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
|
wenzelm@26145
|
964 |
from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
|
bulwahn@26143
|
965 |
with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
|
bulwahn@26143
|
966 |
qed
|
bulwahn@26143
|
967 |
qed
|
bulwahn@26143
|
968 |
qed
|
bulwahn@26143
|
969 |
|
wenzelm@26145
|
970 |
text {*
|
wenzelm@26145
|
971 |
This lemma shows which properties suffice to show that a function
|
wenzelm@26145
|
972 |
@{text "f"} with @{text "f xs = ys"} behaves like sort.
|
wenzelm@26145
|
973 |
*}
|
wenzelm@26145
|
974 |
lemma properties_for_sort:
|
wenzelm@26145
|
975 |
"multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
|
bulwahn@26143
|
976 |
proof (induct xs arbitrary: ys)
|
wenzelm@26145
|
977 |
case Nil then show ?case by simp
|
bulwahn@26143
|
978 |
next
|
bulwahn@26143
|
979 |
case (Cons x xs)
|
wenzelm@26145
|
980 |
then have "x \<in> set ys"
|
wenzelm@26145
|
981 |
by (auto simp add: mem_set_multiset_eq intro!: ccontr)
|
bulwahn@26143
|
982 |
with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
|
bulwahn@26143
|
983 |
by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
|
bulwahn@26143
|
984 |
qed
|
bulwahn@26143
|
985 |
|
kleing@15867
|
986 |
|
paulson@15072
|
987 |
subsection {* Pointwise ordering induced by count *}
|
paulson@15072
|
988 |
|
wenzelm@19086
|
989 |
definition
|
kleing@25610
|
990 |
mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where
|
haftmann@27106
|
991 |
[code func del]: "(A \<le># B) = (\<forall>a. count A a \<le> count B a)"
|
wenzelm@26145
|
992 |
|
nipkow@23611
|
993 |
definition
|
kleing@25610
|
994 |
mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
|
haftmann@27106
|
995 |
[code func del]: "(A <# B) = (A \<le># B \<and> A \<noteq> B)"
|
kleing@25610
|
996 |
|
wenzelm@26145
|
997 |
notation mset_le (infix "\<subseteq>#" 50)
|
wenzelm@26145
|
998 |
notation mset_less (infix "\<subset>#" 50)
|
paulson@15072
|
999 |
|
nipkow@23611
|
1000 |
lemma mset_le_refl[simp]: "A \<le># A"
|
nipkow@26178
|
1001 |
unfolding mset_le_def by auto
|
paulson@15072
|
1002 |
|
wenzelm@26145
|
1003 |
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
|
nipkow@26178
|
1004 |
unfolding mset_le_def by (fast intro: order_trans)
|
paulson@15072
|
1005 |
|
wenzelm@26145
|
1006 |
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
|
nipkow@26178
|
1007 |
apply (unfold mset_le_def)
|
nipkow@26178
|
1008 |
apply (rule multiset_eq_conv_count_eq [THEN iffD2])
|
nipkow@26178
|
1009 |
apply (blast intro: order_antisym)
|
nipkow@26178
|
1010 |
done
|
paulson@15072
|
1011 |
|
wenzelm@26145
|
1012 |
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
|
nipkow@26178
|
1013 |
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
|
nipkow@26178
|
1014 |
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
|
nipkow@26178
|
1015 |
done
|
paulson@15072
|
1016 |
|
nipkow@23611
|
1017 |
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
|
nipkow@26178
|
1018 |
unfolding mset_le_def by auto
|
paulson@15072
|
1019 |
|
nipkow@23611
|
1020 |
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
|
nipkow@26178
|
1021 |
unfolding mset_le_def by auto
|
paulson@15072
|
1022 |
|
nipkow@23611
|
1023 |
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
|
nipkow@26178
|
1024 |
apply (unfold mset_le_def)
|
nipkow@26178
|
1025 |
apply auto
|
nipkow@26178
|
1026 |
apply (erule_tac x = a in allE)+
|
nipkow@26178
|
1027 |
apply auto
|
nipkow@26178
|
1028 |
done
|
paulson@15072
|
1029 |
|
nipkow@23611
|
1030 |
lemma mset_le_add_left[simp]: "A \<le># A + B"
|
nipkow@26178
|
1031 |
unfolding mset_le_def by auto
|
paulson@15072
|
1032 |
|
nipkow@23611
|
1033 |
lemma mset_le_add_right[simp]: "B \<le># A + B"
|
nipkow@26178
|
1034 |
unfolding mset_le_def by auto
|
paulson@15072
|
1035 |
|
bulwahn@26143
|
1036 |
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
|
nipkow@26178
|
1037 |
by (simp add: mset_le_def)
|
bulwahn@26143
|
1038 |
|
bulwahn@26143
|
1039 |
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
|
nipkow@26178
|
1040 |
by (simp add: multiset_eq_conv_count_eq mset_le_def)
|
bulwahn@26143
|
1041 |
|
bulwahn@26143
|
1042 |
lemma mset_le_multiset_union_diff_commute:
|
nipkow@26178
|
1043 |
assumes "B \<le># A"
|
nipkow@26178
|
1044 |
shows "A - B + C = A + C - B"
|
bulwahn@26143
|
1045 |
proof -
|
wenzelm@26145
|
1046 |
from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
|
wenzelm@26145
|
1047 |
from this obtain D where "A = B + D" ..
|
wenzelm@26145
|
1048 |
then show ?thesis
|
wenzelm@26145
|
1049 |
apply simp
|
wenzelm@26145
|
1050 |
apply (subst union_commute)
|
wenzelm@26145
|
1051 |
apply (subst multiset_diff_union_assoc)
|
wenzelm@26145
|
1052 |
apply simp
|
wenzelm@26145
|
1053 |
apply (simp add: diff_cancel)
|
wenzelm@26145
|
1054 |
apply (subst union_assoc)
|
wenzelm@26145
|
1055 |
apply (subst union_commute[of "B" _])
|
wenzelm@26145
|
1056 |
apply (subst multiset_diff_union_assoc)
|
wenzelm@26145
|
1057 |
apply simp
|
wenzelm@26145
|
1058 |
apply (simp add: diff_cancel)
|
wenzelm@26145
|
1059 |
done
|
bulwahn@26143
|
1060 |
qed
|
bulwahn@26143
|
1061 |
|
nipkow@23611
|
1062 |
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
|
nipkow@26178
|
1063 |
apply (induct xs)
|
nipkow@26178
|
1064 |
apply auto
|
nipkow@26178
|
1065 |
apply (rule mset_le_trans)
|
nipkow@26178
|
1066 |
apply auto
|
nipkow@26178
|
1067 |
done
|
nipkow@23611
|
1068 |
|
wenzelm@26145
|
1069 |
lemma multiset_of_update:
|
wenzelm@26145
|
1070 |
"i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
|
bulwahn@26143
|
1071 |
proof (induct ls arbitrary: i)
|
wenzelm@26145
|
1072 |
case Nil then show ?case by simp
|
bulwahn@26143
|
1073 |
next
|
bulwahn@26143
|
1074 |
case (Cons x xs)
|
bulwahn@26143
|
1075 |
show ?case
|
wenzelm@26145
|
1076 |
proof (cases i)
|
wenzelm@26145
|
1077 |
case 0 then show ?thesis by simp
|
wenzelm@26145
|
1078 |
next
|
wenzelm@26145
|
1079 |
case (Suc i')
|
wenzelm@26145
|
1080 |
with Cons show ?thesis
|
wenzelm@26145
|
1081 |
apply simp
|
wenzelm@26145
|
1082 |
apply (subst union_assoc)
|
wenzelm@26145
|
1083 |
apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
|
wenzelm@26145
|
1084 |
apply (subst union_assoc [symmetric])
|
wenzelm@26145
|
1085 |
apply simp
|
wenzelm@26145
|
1086 |
apply (rule mset_le_multiset_union_diff_commute)
|
wenzelm@26145
|
1087 |
apply (simp add: mset_le_single nth_mem_multiset_of)
|
wenzelm@26145
|
1088 |
done
|
bulwahn@26143
|
1089 |
qed
|
bulwahn@26143
|
1090 |
qed
|
bulwahn@26143
|
1091 |
|
wenzelm@26145
|
1092 |
lemma multiset_of_swap:
|
wenzelm@26145
|
1093 |
"i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
|
wenzelm@26145
|
1094 |
multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
|
nipkow@26178
|
1095 |
apply (case_tac "i = j")
|
nipkow@26178
|
1096 |
apply simp
|
nipkow@26178
|
1097 |
apply (simp add: multiset_of_update)
|
nipkow@26178
|
1098 |
apply (subst elem_imp_eq_diff_union[symmetric])
|
nipkow@26178
|
1099 |
apply (simp add: nth_mem_multiset_of)
|
nipkow@26178
|
1100 |
apply simp
|
nipkow@26178
|
1101 |
done
|
bulwahn@26143
|
1102 |
|
wenzelm@26145
|
1103 |
interpretation mset_order: order ["op \<le>#" "op <#"]
|
nipkow@26178
|
1104 |
by (auto intro: order.intro mset_le_refl mset_le_antisym
|
haftmann@25208
|
1105 |
mset_le_trans simp: mset_less_def)
|
nipkow@23611
|
1106 |
|
nipkow@23611
|
1107 |
interpretation mset_order_cancel_semigroup:
|
wenzelm@26145
|
1108 |
pordered_cancel_ab_semigroup_add ["op +" "op \<le>#" "op <#"]
|
nipkow@26178
|
1109 |
by unfold_locales (erule mset_le_mono_add [OF mset_le_refl])
|
nipkow@23611
|
1110 |
|
nipkow@23611
|
1111 |
interpretation mset_order_semigroup_cancel:
|
wenzelm@26145
|
1112 |
pordered_ab_semigroup_add_imp_le ["op +" "op \<le>#" "op <#"]
|
nipkow@26178
|
1113 |
by (unfold_locales) simp
|
paulson@15072
|
1114 |
|
kleing@25610
|
1115 |
|
wenzelm@26145
|
1116 |
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
|
nipkow@26178
|
1117 |
apply (clarsimp simp: mset_le_def mset_less_def)
|
nipkow@26178
|
1118 |
apply (erule_tac x=x in allE)
|
nipkow@26178
|
1119 |
apply auto
|
nipkow@26178
|
1120 |
done
|
kleing@25610
|
1121 |
|
wenzelm@26145
|
1122 |
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
|
nipkow@26178
|
1123 |
apply (clarsimp simp: mset_le_def mset_less_def)
|
nipkow@26178
|
1124 |
apply (erule_tac x = x in allE)
|
nipkow@26178
|
1125 |
apply auto
|
nipkow@26178
|
1126 |
done
|
kleing@25610
|
1127 |
|
wenzelm@26145
|
1128 |
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
|
nipkow@26178
|
1129 |
apply (rule conjI)
|
nipkow@26178
|
1130 |
apply (simp add: mset_lessD)
|
nipkow@26178
|
1131 |
apply (clarsimp simp: mset_le_def mset_less_def)
|
nipkow@26178
|
1132 |
apply safe
|
nipkow@26178
|
1133 |
apply (erule_tac x = a in allE)
|
nipkow@26178
|
1134 |
apply (auto split: split_if_asm)
|
nipkow@26178
|
1135 |
done
|
kleing@25610
|
1136 |
|
wenzelm@26145
|
1137 |
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
|
nipkow@26178
|
1138 |
apply (rule conjI)
|
nipkow@26178
|
1139 |
apply (simp add: mset_leD)
|
nipkow@26178
|
1140 |
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
|
nipkow@26178
|
1141 |
done
|
kleing@25610
|
1142 |
|
kleing@25610
|
1143 |
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False"
|
nipkow@26178
|
1144 |
by (induct A) (auto simp: mset_le_def mset_less_def)
|
kleing@25610
|
1145 |
|
kleing@25610
|
1146 |
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
|
nipkow@26178
|
1147 |
by (auto simp: mset_le_def mset_less_def)
|
kleing@25610
|
1148 |
|
kleing@25610
|
1149 |
lemma multi_psub_self[simp]: "A \<subset># A = False"
|
nipkow@26178
|
1150 |
by (auto simp: mset_le_def mset_less_def)
|
kleing@25610
|
1151 |
|
kleing@25610
|
1152 |
lemma mset_less_add_bothsides:
|
kleing@25610
|
1153 |
"T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
|
nipkow@26178
|
1154 |
by (auto simp: mset_le_def mset_less_def)
|
kleing@25610
|
1155 |
|
kleing@25610
|
1156 |
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
|
nipkow@26178
|
1157 |
by (auto simp: mset_le_def mset_less_def)
|
kleing@25610
|
1158 |
|
kleing@25610
|
1159 |
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
|
kleing@25610
|
1160 |
proof (induct A arbitrary: B)
|
kleing@25610
|
1161 |
case (empty M)
|
wenzelm@26145
|
1162 |
then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
|
kleing@25610
|
1163 |
then obtain M' x where "M = M' + {#x#}"
|
kleing@25610
|
1164 |
by (blast dest: multi_nonempty_split)
|
wenzelm@26145
|
1165 |
then show ?case by simp
|
kleing@25610
|
1166 |
next
|
kleing@25610
|
1167 |
case (add S x T)
|
kleing@25610
|
1168 |
have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
|
kleing@25610
|
1169 |
have SxsubT: "S + {#x#} \<subset># T" by fact
|
wenzelm@26145
|
1170 |
then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
|
kleing@25610
|
1171 |
then obtain T' where T: "T = T' + {#x#}"
|
kleing@25610
|
1172 |
by (blast dest: multi_member_split)
|
wenzelm@26145
|
1173 |
then have "S \<subset># T'" using SxsubT
|
kleing@25610
|
1174 |
by (blast intro: mset_less_add_bothsides)
|
wenzelm@26145
|
1175 |
then have "size S < size T'" using IH by simp
|
wenzelm@26145
|
1176 |
then show ?case using T by simp
|
kleing@25610
|
1177 |
qed
|
kleing@25610
|
1178 |
|
kleing@25610
|
1179 |
lemmas mset_less_trans = mset_order.less_eq_less.less_trans
|
kleing@25610
|
1180 |
|
kleing@25610
|
1181 |
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
|
nipkow@26178
|
1182 |
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
|
kleing@25610
|
1183 |
|
wenzelm@26145
|
1184 |
|
kleing@25610
|
1185 |
subsection {* Strong induction and subset induction for multisets *}
|
kleing@25610
|
1186 |
|
nipkow@26016
|
1187 |
text {* Well-foundedness of proper subset operator: *}
|
kleing@25610
|
1188 |
|
wenzelm@26145
|
1189 |
text {* proper multiset subset *}
|
kleing@25610
|
1190 |
definition
|
wenzelm@26145
|
1191 |
mset_less_rel :: "('a multiset * 'a multiset) set" where
|
wenzelm@26145
|
1192 |
"mset_less_rel = {(A,B). A \<subset># B}"
|
kleing@25610
|
1193 |
|
kleing@25610
|
1194 |
lemma multiset_add_sub_el_shuffle:
|
wenzelm@26145
|
1195 |
assumes "c \<in># B" and "b \<noteq> c"
|
kleing@25610
|
1196 |
shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
|
kleing@25610
|
1197 |
proof -
|
wenzelm@26145
|
1198 |
from `c \<in># B` obtain A where B: "B = A + {#c#}"
|
kleing@25610
|
1199 |
by (blast dest: multi_member_split)
|
kleing@25610
|
1200 |
have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
|
wenzelm@26145
|
1201 |
then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
|
kleing@25610
|
1202 |
by (simp add: union_ac)
|
wenzelm@26145
|
1203 |
then show ?thesis using B by simp
|
kleing@25610
|
1204 |
qed
|
kleing@25610
|
1205 |
|
kleing@25610
|
1206 |
lemma wf_mset_less_rel: "wf mset_less_rel"
|
nipkow@26178
|
1207 |
apply (unfold mset_less_rel_def)
|
nipkow@26178
|
1208 |
apply (rule wf_measure [THEN wf_subset, where f1=size])
|
nipkow@26178
|
1209 |
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
|
nipkow@26178
|
1210 |
done
|
kleing@25610
|
1211 |
|
nipkow@26016
|
1212 |
text {* The induction rules: *}
|
kleing@25610
|
1213 |
|
kleing@25610
|
1214 |
lemma full_multiset_induct [case_names less]:
|
nipkow@26178
|
1215 |
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
|
nipkow@26178
|
1216 |
shows "P B"
|
nipkow@26178
|
1217 |
apply (rule wf_mset_less_rel [THEN wf_induct])
|
nipkow@26178
|
1218 |
apply (rule ih, auto simp: mset_less_rel_def)
|
nipkow@26178
|
1219 |
done
|
kleing@25610
|
1220 |
|
kleing@25610
|
1221 |
lemma multi_subset_induct [consumes 2, case_names empty add]:
|
nipkow@26178
|
1222 |
assumes "F \<subseteq># A"
|
nipkow@26178
|
1223 |
and empty: "P {#}"
|
nipkow@26178
|
1224 |
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
|
nipkow@26178
|
1225 |
shows "P F"
|
kleing@25610
|
1226 |
proof -
|
kleing@25610
|
1227 |
from `F \<subseteq># A`
|
kleing@25610
|
1228 |
show ?thesis
|
kleing@25610
|
1229 |
proof (induct F)
|
kleing@25610
|
1230 |
show "P {#}" by fact
|
kleing@25610
|
1231 |
next
|
kleing@25610
|
1232 |
fix x F
|
kleing@25610
|
1233 |
assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
|
kleing@25610
|
1234 |
show "P (F + {#x#})"
|
kleing@25610
|
1235 |
proof (rule insert)
|
kleing@25610
|
1236 |
from i show "x \<in># A" by (auto dest: mset_le_insertD)
|
wenzelm@26145
|
1237 |
from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
|
kleing@25610
|
1238 |
with P show "P F" .
|
kleing@25610
|
1239 |
qed
|
kleing@25610
|
1240 |
qed
|
kleing@25610
|
1241 |
qed
|
kleing@25610
|
1242 |
|
nipkow@26016
|
1243 |
text{* A consequence: Extensionality. *}
|
kleing@25610
|
1244 |
|
wenzelm@26145
|
1245 |
lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
|
nipkow@26178
|
1246 |
apply (rule iffI)
|
nipkow@26178
|
1247 |
prefer 2
|
nipkow@26178
|
1248 |
apply clarsimp
|
nipkow@26178
|
1249 |
apply (induct A arbitrary: B rule: full_multiset_induct)
|
nipkow@26178
|
1250 |
apply (rename_tac C)
|
nipkow@26178
|
1251 |
apply (case_tac B rule: multiset_cases)
|
nipkow@26178
|
1252 |
apply (simp add: empty_multiset_count)
|
nipkow@26178
|
1253 |
apply simp
|
nipkow@26178
|
1254 |
apply (case_tac "x \<in># C")
|
nipkow@26178
|
1255 |
apply (force dest: multi_member_split)
|
nipkow@26178
|
1256 |
apply (erule_tac x = x in allE)
|
nipkow@26178
|
1257 |
apply simp
|
nipkow@26178
|
1258 |
done
|
kleing@25610
|
1259 |
|
kleing@25610
|
1260 |
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
|
kleing@25610
|
1261 |
|
wenzelm@26145
|
1262 |
|
kleing@25610
|
1263 |
subsection {* The fold combinator *}
|
kleing@25610
|
1264 |
|
wenzelm@26145
|
1265 |
text {*
|
wenzelm@26145
|
1266 |
The intended behaviour is
|
wenzelm@26145
|
1267 |
@{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
|
wenzelm@26145
|
1268 |
if @{text f} is associative-commutative.
|
kleing@25610
|
1269 |
*}
|
kleing@25610
|
1270 |
|
wenzelm@26145
|
1271 |
text {*
|
wenzelm@26145
|
1272 |
The graph of @{text "fold_mset"}, @{text "z"}: the start element,
|
wenzelm@26145
|
1273 |
@{text "f"}: folding function, @{text "A"}: the multiset, @{text
|
wenzelm@26145
|
1274 |
"y"}: the result.
|
wenzelm@26145
|
1275 |
*}
|
kleing@25610
|
1276 |
inductive
|
kleing@25759
|
1277 |
fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool"
|
kleing@25610
|
1278 |
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
|
kleing@25610
|
1279 |
and z :: 'b
|
kleing@25610
|
1280 |
where
|
kleing@25759
|
1281 |
emptyI [intro]: "fold_msetG f z {#} z"
|
kleing@25759
|
1282 |
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
|
kleing@25610
|
1283 |
|
kleing@25759
|
1284 |
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
|
kleing@25759
|
1285 |
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y"
|
kleing@25610
|
1286 |
|
kleing@25610
|
1287 |
definition
|
wenzelm@26145
|
1288 |
fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
|
wenzelm@26145
|
1289 |
"fold_mset f z A = (THE x. fold_msetG f z A x)"
|
kleing@25610
|
1290 |
|
kleing@25759
|
1291 |
lemma Diff1_fold_msetG:
|
wenzelm@26145
|
1292 |
"fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
|
nipkow@26178
|
1293 |
apply (frule_tac x = x in fold_msetG.insertI)
|
nipkow@26178
|
1294 |
apply auto
|
nipkow@26178
|
1295 |
done
|
kleing@25610
|
1296 |
|
kleing@25759
|
1297 |
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
|
nipkow@26178
|
1298 |
apply (induct A)
|
nipkow@26178
|
1299 |
apply blast
|
nipkow@26178
|
1300 |
apply clarsimp
|
nipkow@26178
|
1301 |
apply (drule_tac x = x in fold_msetG.insertI)
|
nipkow@26178
|
1302 |
apply auto
|
nipkow@26178
|
1303 |
done
|
kleing@25610
|
1304 |
|
kleing@25759
|
1305 |
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
|
nipkow@26178
|
1306 |
unfolding fold_mset_def by blast
|
kleing@25610
|
1307 |
|
kleing@25610
|
1308 |
locale left_commutative =
|
nipkow@26178
|
1309 |
fixes f :: "'a => 'b => 'b"
|
nipkow@26178
|
1310 |
assumes left_commute: "f x (f y z) = f y (f x z)"
|
wenzelm@26145
|
1311 |
begin
|
kleing@25610
|
1312 |
|
wenzelm@26145
|
1313 |
lemma fold_msetG_determ:
|
wenzelm@26145
|
1314 |
"fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
|
kleing@25610
|
1315 |
proof (induct arbitrary: x y z rule: full_multiset_induct)
|
kleing@25610
|
1316 |
case (less M x\<^isub>1 x\<^isub>2 Z)
|
kleing@25610
|
1317 |
have IH: "\<forall>A. A \<subset># M \<longrightarrow>
|
kleing@25759
|
1318 |
(\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
|
kleing@25610
|
1319 |
\<longrightarrow> x' = x)" by fact
|
kleing@25759
|
1320 |
have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+
|
kleing@25610
|
1321 |
show ?case
|
kleing@25759
|
1322 |
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
|
kleing@25610
|
1323 |
assume "M = {#}" and "x\<^isub>1 = Z"
|
wenzelm@26145
|
1324 |
then show ?case using Mfoldx\<^isub>2 by auto
|
kleing@25610
|
1325 |
next
|
kleing@25610
|
1326 |
fix B b u
|
kleing@25759
|
1327 |
assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
|
wenzelm@26145
|
1328 |
then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
|
kleing@25610
|
1329 |
show ?case
|
kleing@25759
|
1330 |
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
|
kleing@25610
|
1331 |
assume "M = {#}" "x\<^isub>2 = Z"
|
wenzelm@26145
|
1332 |
then show ?case using Mfoldx\<^isub>1 by auto
|
kleing@25610
|
1333 |
next
|
kleing@25610
|
1334 |
fix C c v
|
kleing@25759
|
1335 |
assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
|
wenzelm@26145
|
1336 |
then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
|
wenzelm@26145
|
1337 |
then have CsubM: "C \<subset># M" by simp
|
kleing@25610
|
1338 |
from MBb have BsubM: "B \<subset># M" by simp
|
kleing@25610
|
1339 |
show ?case
|
kleing@25610
|
1340 |
proof cases
|
kleing@25610
|
1341 |
assume "b=c"
|
kleing@25610
|
1342 |
then moreover have "B = C" using MBb MCc by auto
|
kleing@25610
|
1343 |
ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
|
kleing@25610
|
1344 |
next
|
kleing@25610
|
1345 |
assume diff: "b \<noteq> c"
|
kleing@25610
|
1346 |
let ?D = "B - {#c#}"
|
kleing@25610
|
1347 |
have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
|
kleing@25610
|
1348 |
by (auto intro: insert_noteq_member dest: sym)
|
kleing@25610
|
1349 |
have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
|
wenzelm@26145
|
1350 |
then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
|
kleing@25610
|
1351 |
from MBb MCc have "B + {#b#} = C + {#c#}" by blast
|
wenzelm@26145
|
1352 |
then have [simp]: "B + {#b#} - {#c#} = C"
|
kleing@25610
|
1353 |
using MBb MCc binC cinB by auto
|
kleing@25610
|
1354 |
have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
|
kleing@25610
|
1355 |
using MBb MCc diff binC cinB
|
kleing@25610
|
1356 |
by (auto simp: multiset_add_sub_el_shuffle)
|
kleing@25759
|
1357 |
then obtain d where Dfoldd: "fold_msetG f Z ?D d"
|
kleing@25759
|
1358 |
using fold_msetG_nonempty by iprover
|
wenzelm@26145
|
1359 |
then have "fold_msetG f Z B (f c d)" using cinB
|
kleing@25759
|
1360 |
by (rule Diff1_fold_msetG)
|
wenzelm@26145
|
1361 |
then have "f c d = u" using IH BsubM Bu by blast
|
kleing@25610
|
1362 |
moreover
|
kleing@25759
|
1363 |
have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
|
kleing@25610
|
1364 |
by (auto simp: multiset_add_sub_el_shuffle
|
kleing@25759
|
1365 |
dest: fold_msetG.insertI [where x=b])
|
wenzelm@26145
|
1366 |
then have "f b d = v" using IH CsubM Cv by blast
|
kleing@25610
|
1367 |
ultimately show ?thesis using x\<^isub>1 x\<^isub>2
|
kleing@25610
|
1368 |
by (auto simp: left_commute)
|
kleing@25610
|
1369 |
qed
|
kleing@25610
|
1370 |
qed
|
kleing@25610
|
1371 |
qed
|
kleing@25610
|
1372 |
qed
|
kleing@25610
|
1373 |
|
wenzelm@26145
|
1374 |
lemma fold_mset_insert_aux:
|
wenzelm@26145
|
1375 |
"(fold_msetG f z (A + {#x#}) v) =
|
kleing@25759
|
1376 |
(\<exists>y. fold_msetG f z A y \<and> v = f x y)"
|
nipkow@26178
|
1377 |
apply (rule iffI)
|
nipkow@26178
|
1378 |
prefer 2
|
nipkow@26178
|
1379 |
apply blast
|
nipkow@26178
|
1380 |
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
|
nipkow@26178
|
1381 |
apply (blast intro: fold_msetG_determ)
|
nipkow@26178
|
1382 |
done
|
kleing@25610
|
1383 |
|
wenzelm@26145
|
1384 |
lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
|
nipkow@26178
|
1385 |
unfolding fold_mset_def by (blast intro: fold_msetG_determ)
|
kleing@25610
|
1386 |
|
wenzelm@26145
|
1387 |
lemma fold_mset_insert:
|
nipkow@26178
|
1388 |
"fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
|
nipkow@26178
|
1389 |
apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)
|
nipkow@26178
|
1390 |
apply (rule the_equality)
|
nipkow@26178
|
1391 |
apply (auto cong add: conj_cong
|
wenzelm@26145
|
1392 |
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
|
nipkow@26178
|
1393 |
done
|
kleing@25759
|
1394 |
|
wenzelm@26145
|
1395 |
lemma fold_mset_insert_idem:
|
nipkow@26178
|
1396 |
"fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"
|
nipkow@26178
|
1397 |
apply (simp add: fold_mset_def fold_mset_insert_aux)
|
nipkow@26178
|
1398 |
apply (rule the_equality)
|
nipkow@26178
|
1399 |
apply (auto cong add: conj_cong
|
wenzelm@26145
|
1400 |
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
|
nipkow@26178
|
1401 |
done
|
kleing@25610
|
1402 |
|
wenzelm@26145
|
1403 |
lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
|
nipkow@26178
|
1404 |
by (induct A) (auto simp: fold_mset_insert left_commute [of x])
|
nipkow@26178
|
1405 |
|
wenzelm@26145
|
1406 |
lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
|
nipkow@26178
|
1407 |
using fold_mset_insert [of z "{#}"] by simp
|
kleing@25610
|
1408 |
|
wenzelm@26145
|
1409 |
lemma fold_mset_union [simp]:
|
wenzelm@26145
|
1410 |
"fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
|
kleing@25759
|
1411 |
proof (induct A)
|
wenzelm@26145
|
1412 |
case empty then show ?case by simp
|
kleing@25759
|
1413 |
next
|
wenzelm@26145
|
1414 |
case (add A x)
|
wenzelm@26145
|
1415 |
have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
|
wenzelm@26145
|
1416 |
then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))"
|
wenzelm@26145
|
1417 |
by (simp add: fold_mset_insert)
|
wenzelm@26145
|
1418 |
also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
|
wenzelm@26145
|
1419 |
by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
|
wenzelm@26145
|
1420 |
finally show ?case .
|
kleing@25759
|
1421 |
qed
|
kleing@25759
|
1422 |
|
wenzelm@26145
|
1423 |
lemma fold_mset_fusion:
|
ballarin@27611
|
1424 |
assumes "left_commutative g"
|
ballarin@27611
|
1425 |
shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
|
ballarin@27611
|
1426 |
proof -
|
ballarin@27611
|
1427 |
interpret left_commutative [g] by fact
|
ballarin@27611
|
1428 |
show "PROP ?P" by (induct A) auto
|
ballarin@27611
|
1429 |
qed
|
kleing@25610
|
1430 |
|
wenzelm@26145
|
1431 |
lemma fold_mset_rec:
|
wenzelm@26145
|
1432 |
assumes "a \<in># A"
|
kleing@25759
|
1433 |
shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
|
kleing@25610
|
1434 |
proof -
|
wenzelm@26145
|
1435 |
from assms obtain A' where "A = A' + {#a#}"
|
wenzelm@26145
|
1436 |
by (blast dest: multi_member_split)
|
wenzelm@26145
|
1437 |
then show ?thesis by simp
|
kleing@25610
|
1438 |
qed
|
kleing@25610
|
1439 |
|
wenzelm@26145
|
1440 |
end
|
wenzelm@26145
|
1441 |
|
wenzelm@26145
|
1442 |
text {*
|
wenzelm@26145
|
1443 |
A note on code generation: When defining some function containing a
|
wenzelm@26145
|
1444 |
subterm @{term"fold_mset F"}, code generation is not automatic. When
|
wenzelm@26145
|
1445 |
interpreting locale @{text left_commutative} with @{text F}, the
|
wenzelm@26145
|
1446 |
would be code thms for @{const fold_mset} become thms like
|
wenzelm@26145
|
1447 |
@{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
|
wenzelm@26145
|
1448 |
contains defined symbols, i.e.\ is not a code thm. Hence a separate
|
wenzelm@26145
|
1449 |
constant with its own code thms needs to be introduced for @{text
|
wenzelm@26145
|
1450 |
F}. See the image operator below.
|
wenzelm@26145
|
1451 |
*}
|
wenzelm@26145
|
1452 |
|
nipkow@26016
|
1453 |
|
nipkow@26016
|
1454 |
subsection {* Image *}
|
nipkow@26016
|
1455 |
|
nipkow@26016
|
1456 |
definition [code func del]: "image_mset f == fold_mset (op + o single o f) {#}"
|
nipkow@26016
|
1457 |
|
wenzelm@26145
|
1458 |
interpretation image_left_comm: left_commutative ["op + o single o f"]
|
nipkow@26178
|
1459 |
by (unfold_locales) (simp add:union_ac)
|
nipkow@26016
|
1460 |
|
wenzelm@26145
|
1461 |
lemma image_mset_empty [simp, code func]: "image_mset f {#} = {#}"
|
nipkow@26178
|
1462 |
by (simp add: image_mset_def)
|
nipkow@26016
|
1463 |
|
wenzelm@26145
|
1464 |
lemma image_mset_single [simp, code func]: "image_mset f {#x#} = {#f x#}"
|
nipkow@26178
|
1465 |
by (simp add: image_mset_def)
|
nipkow@26016
|
1466 |
|
nipkow@26016
|
1467 |
lemma image_mset_insert:
|
nipkow@26016
|
1468 |
"image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
|
nipkow@26178
|
1469 |
by (simp add: image_mset_def add_ac)
|
nipkow@26016
|
1470 |
|
nipkow@26016
|
1471 |
lemma image_mset_union[simp, code func]:
|
nipkow@26016
|
1472 |
"image_mset f (M+N) = image_mset f M + image_mset f N"
|
nipkow@26178
|
1473 |
apply (induct N)
|
nipkow@26178
|
1474 |
apply simp
|
nipkow@26178
|
1475 |
apply (simp add: union_assoc [symmetric] image_mset_insert)
|
nipkow@26178
|
1476 |
done
|
nipkow@26016
|
1477 |
|
wenzelm@26145
|
1478 |
lemma size_image_mset [simp]: "size (image_mset f M) = size M"
|
nipkow@26178
|
1479 |
by (induct M) simp_all
|
nipkow@26016
|
1480 |
|
wenzelm@26145
|
1481 |
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
|
nipkow@26178
|
1482 |
by (cases M) auto
|
nipkow@26016
|
1483 |
|
nipkow@26016
|
1484 |
|
wenzelm@26145
|
1485 |
syntax
|
wenzelm@26145
|
1486 |
comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
|
wenzelm@26145
|
1487 |
("({#_/. _ :# _#})")
|
wenzelm@26145
|
1488 |
translations
|
wenzelm@26145
|
1489 |
"{#e. x:#M#}" == "CONST image_mset (%x. e) M"
|
nipkow@26016
|
1490 |
|
wenzelm@26145
|
1491 |
syntax
|
wenzelm@26145
|
1492 |
comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
|
wenzelm@26145
|
1493 |
("({#_/ | _ :# _./ _#})")
|
nipkow@26016
|
1494 |
translations
|
nipkow@26033
|
1495 |
"{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
|
nipkow@26016
|
1496 |
|
wenzelm@26145
|
1497 |
text {*
|
wenzelm@26145
|
1498 |
This allows to write not just filters like @{term "{#x:#M. x<c#}"}
|
wenzelm@26145
|
1499 |
but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
|
wenzelm@26145
|
1500 |
"{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
|
wenzelm@26145
|
1501 |
@{term "{#x+x|x:#M. x<c#}"}.
|
wenzelm@26145
|
1502 |
*}
|
nipkow@26016
|
1503 |
|
wenzelm@10249
|
1504 |
end
|