author  haftmann 
Fri, 22 Jan 2010 13:38:28 +0100  
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permissions  rwrr 
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(* Title: HOL/Library/Multiset.thy 
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Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker 
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*) 
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header {* (Finite) multisets *} 
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theory Multiset 
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imports Main 
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begin 
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subsection {* The type of multisets *} 

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typedef 'a multiset = "{f :: 'a => nat. finite {x. f x > 0}}" 
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morphisms count Abs_multiset 
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proof 
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show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp 
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qed 
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lemmas multiset_typedef = Abs_multiset_inverse count_inverse count 
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abbreviation Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where 
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"a :# M == 0 < count M a" 
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notation (xsymbols) 
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Melem (infix "\<in>#" 50) 

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lemma multiset_eq_conv_count_eq: 
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"M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)" 
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by (simp only: count_inject [symmetric] expand_fun_eq) 
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lemma multi_count_ext: 
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"(\<And>x. count A x = count B x) \<Longrightarrow> A = B" 
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using multiset_eq_conv_count_eq by auto 
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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: 
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"(\<lambda>a. 0) \<in> multiset" 
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by (simp add: multiset_def) 
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lemma only1_in_multiset: 
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"(\<lambda>b. if b = a then n else 0) \<in> multiset" 
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by (simp add: multiset_def) 
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lemma union_preserves_multiset: 
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"M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset" 
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by (simp add: multiset_def) 
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lemma diff_preserves_multiset: 
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assumes "M \<in> multiset" 
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shows "(\<lambda>a. M a  N a) \<in> multiset" 
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proof  
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have "{x. N x < M x} \<subseteq> {x. 0 < M x}" 
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by auto 
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with assms show ?thesis 
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by (auto simp add: multiset_def intro: finite_subset) 
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qed 
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lemma MCollect_preserves_multiset: 
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assumes "M \<in> multiset" 
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shows "(\<lambda>x. if P x then M x else 0) \<in> multiset" 
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proof  
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have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}" 
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by auto 
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with assms show ?thesis 
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by (auto simp add: multiset_def intro: finite_subset) 
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qed 
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lemmas in_multiset = const0_in_multiset only1_in_multiset 
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union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset 
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subsection {* Representing multisets *} 
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text {* Multiset comprehension *} 
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definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where 
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"MCollect M P = Abs_multiset (\<lambda>x. if P x then count M x else 0)" 
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syntax 
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"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ :# _./ _#})") 
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translations 
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"{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)" 
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text {* Multiset enumeration *} 
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instantiation multiset :: (type) "{zero, plus}" 
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begin 
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definition Mempty_def: 
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"0 = Abs_multiset (\<lambda>a. 0)" 
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abbreviation Mempty :: "'a multiset" ("{#}") where 
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"Mempty \<equiv> 0" 
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definition union_def: 
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"M + N = Abs_multiset (\<lambda>a. count M a + count N a)" 
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instance .. 
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end 
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definition single :: "'a => 'a multiset" where 
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"single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)" 
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syntax 
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"_multiset" :: "args => 'a multiset" ("{#(_)#}") 
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translations 
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"{#x, xs#}" == "{#x#} + {#xs#}" 

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"{#x#}" == "CONST single x" 

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lemma count_empty [simp]: "count {#} a = 0" 
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by (simp add: Mempty_def in_multiset multiset_typedef) 
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)" 
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by (simp add: single_def in_multiset multiset_typedef) 
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subsection {* Basic operations *} 
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subsubsection {* Union *} 

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lemma count_union [simp]: "count (M + N) a = count M a + count N a" 
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by (simp add: union_def in_multiset multiset_typedef) 
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instance multiset :: (type) cancel_comm_monoid_add proof 
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qed (simp_all add: multiset_eq_conv_count_eq) 
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subsubsection {* Difference *} 

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instantiation multiset :: (type) minus 
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begin 
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definition diff_def: 
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"M  N = Abs_multiset (\<lambda>a. count M a  count N a)" 
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141 
instance .. 
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end 
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lemma count_diff [simp]: "count (M  N) a = count M a  count N a" 
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by (simp add: diff_def in_multiset multiset_typedef) 
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lemma diff_empty [simp]: "M  {#} = M \<and> {#}  M = {#}" 
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by (simp add: Mempty_def diff_def in_multiset multiset_typedef) 
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lemma diff_union_inverse2 [simp]: "M + {#a#}  {#a#} = M" 
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by (rule multi_count_ext) 
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(auto simp del: count_single simp add: union_def diff_def in_multiset multiset_typedef) 
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lemma diff_cancel: "A  A = {#}" 
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by (rule multi_count_ext) simp 
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lemma insert_DiffM: 
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"x \<in># M \<Longrightarrow> {#x#} + (M  {#x#}) = M" 
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by (clarsimp simp: multiset_eq_conv_count_eq) 
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lemma insert_DiffM2 [simp]: 
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"x \<in># M \<Longrightarrow> M  {#x#} + {#x#} = M" 
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by (clarsimp simp: multiset_eq_conv_count_eq) 
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lemma diff_right_commute: 
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"(M::'a multiset)  N  Q = M  Q  N" 
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by (auto simp add: multiset_eq_conv_count_eq) 
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lemma diff_union_swap: 
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"a \<noteq> b \<Longrightarrow> M  {#a#} + {#b#} = M + {#b#}  {#a#}" 
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by (auto simp add: multiset_eq_conv_count_eq) 
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lemma diff_union_single_conv: 
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"a \<in># J \<Longrightarrow> I + J  {#a#} = I + (J  {#a#})" 
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by (simp add: multiset_eq_conv_count_eq) 
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subsubsection {* Intersection *} 
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definition multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where 
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"multiset_inter A B = A  (A  B)" 
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lemma multiset_inter_count: 
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"count (A #\<inter> B) x = min (count A x) (count B x)" 
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by (simp add: multiset_inter_def multiset_typedef) 
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lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A" 
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by (rule multi_count_ext) (simp add: multiset_inter_count) 
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lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C" 
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by (rule multi_count_ext) (simp add: multiset_inter_count) 
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lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)" 
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by (rule multi_count_ext) (simp add: multiset_inter_count) 
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lemmas multiset_inter_ac = 
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multiset_inter_commute 
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multiset_inter_assoc 
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multiset_inter_left_commute 
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lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}" 
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by (rule multi_count_ext) (auto simp add: multiset_inter_count) 
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lemma multiset_union_diff_commute: 
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assumes "B #\<inter> C = {#}" 
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shows "A + B  C = A  C + B" 
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proof (rule multi_count_ext) 
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fix x 
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from assms have "min (count B x) (count C x) = 0" 
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by (auto simp add: multiset_inter_count multiset_eq_conv_count_eq) 
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then have "count B x = 0 \<or> count C x = 0" 
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by auto 
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then show "count (A + B  C) x = count (A  C + B) x" 
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by auto 
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qed 
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subsubsection {* Comprehension (filter) *} 
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221 
lemma count_MCollect [simp]: 

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"count {# x:#M. P x #} a = (if P a then count M a else 0)" 
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by (simp add: MCollect_def in_multiset multiset_typedef) 
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224 

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lemma MCollect_empty [simp]: "MCollect {#} P = {#}" 
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by (rule multi_count_ext) simp 
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lemma MCollect_single [simp]: 
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"MCollect {#x#} P = (if P x then {#x#} else {#})" 
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by (rule multi_count_ext) simp 
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lemma MCollect_union [simp]: 
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"MCollect (M + N) f = MCollect M f + MCollect N f" 
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by (rule multi_count_ext) simp 
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235 

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236 

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subsubsection {* Equality of multisets *} 
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}" 
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by (simp add: multiset_eq_conv_count_eq) 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

241 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

242 
lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

243 
by (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

244 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

245 
lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

246 
by (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

247 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

248 
lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

249 
by (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

250 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset

251 
lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

252 
by (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

253 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

254 
lemma diff_single_trivial: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

255 
"\<not> x \<in># M \<Longrightarrow> M  {#x#} = M" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

256 
by (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

257 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

258 
lemma diff_single_eq_union: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

259 
"x \<in># M \<Longrightarrow> M  {#x#} = N \<longleftrightarrow> M = N + {#x#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

260 
by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

261 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

262 
lemma union_single_eq_diff: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

263 
"M + {#x#} = N \<Longrightarrow> M = N  {#x#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

264 
by (auto dest: sym) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

265 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

266 
lemma union_single_eq_member: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

267 
"M + {#x#} = N \<Longrightarrow> x \<in># N" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

268 
by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

269 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

270 
lemma union_is_single: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

271 
"M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs") 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

272 
proof 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

273 
assume ?rhs then show ?lhs by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

274 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

275 
assume ?lhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

276 
then have "\<And>b. count (M + N) b = (if b = a then 1 else 0)" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

277 
then have *: "\<And>b. count M b + count N b = (if b = a then 1 else 0)" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

278 
then have "count M a + count N a = 1" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

279 
then have **: "count M a = 1 \<and> count N a = 0 \<or> count M a = 0 \<and> count N a = 1" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

280 
by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

281 
from * have "\<And>b. b \<noteq> a \<Longrightarrow> count M b + count N b = 0" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

282 
then have ***: "\<And>b. b \<noteq> a \<Longrightarrow> count M b = 0 \<and> count N b = 0" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

283 
from ** and *** have 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

284 
"(\<forall>b. count M b = (if b = a then 1 else 0) \<and> count N b = 0) \<or> 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

285 
(\<forall>b. count M b = 0 \<and> count N b = (if b = a then 1 else 0))" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

286 
by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

287 
then have 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

288 
"(\<forall>b. count M b = (if b = a then 1 else 0)) \<and> (\<forall>b. count N b = 0) \<or> 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

289 
(\<forall>b. count M b = 0) \<and> (\<forall>b. count N b = (if b = a then 1 else 0))" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

290 
by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

291 
then show ?rhs by (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

292 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

293 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

294 
lemma single_is_union: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

295 
"{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

296 
by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

297 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

298 
lemma add_eq_conv_diff: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

299 
"M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N  {#a#} + {#b#} \<and> N = M  {#b#} + {#a#}" (is "?lhs = ?rhs") 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

300 
proof 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

301 
assume ?rhs then show ?lhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

302 
by (auto simp add: add_assoc add_commute [of "{#b#}"]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

303 
(drule sym, simp add: add_assoc [symmetric]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

304 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

305 
assume ?lhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

306 
show ?rhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

307 
proof (cases "a = b") 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

308 
case True with `?lhs` show ?thesis by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

309 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

310 
case False 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

311 
from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

312 
with False have "a \<in># N" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

313 
moreover from `?lhs` have "M = N + {#b#}  {#a#}" by (rule union_single_eq_diff) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

314 
moreover note False 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

315 
ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

316 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

317 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

318 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

319 
lemma insert_noteq_member: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

320 
assumes BC: "B + {#b#} = C + {#c#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

321 
and bnotc: "b \<noteq> c" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

322 
shows "c \<in># B" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

323 
proof  
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

324 
have "c \<in># C + {#c#}" by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

325 
have nc: "\<not> c \<in># {#b#}" using bnotc by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

326 
then have "c \<in># B + {#b#}" using BC by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

327 
then show "c \<in># B" using nc by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

328 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

329 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

330 
lemma add_eq_conv_ex: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

331 
"(M + {#a#} = N + {#b#}) = 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

332 
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))" 
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333 
by (auto simp add: add_eq_conv_diff) 
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334 

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335 

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336 
subsubsection {* Pointwise ordering induced by count *} 
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337 

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338 
definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where 
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339 
"A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)" 
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340 

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341 
definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where 
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342 
"A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B" 
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343 

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344 
notation mset_le (infix "\<subseteq>#" 50) 
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345 
notation mset_less (infix "\<subset>#" 50) 
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346 

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347 
lemma mset_less_eqI: 
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348 
"(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<subseteq># B" 
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349 
by (simp add: mset_le_def) 
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350 

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351 
lemma mset_le_refl[simp]: "A \<le># A" 
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352 
unfolding mset_le_def by auto 
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353 

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354 
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C" 
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355 
unfolding mset_le_def by (fast intro: order_trans) 
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356 

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357 
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B" 
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358 
apply (unfold mset_le_def) 
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359 
apply (rule multiset_eq_conv_count_eq [THEN iffD2]) 
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360 
apply (blast intro: order_antisym) 
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361 
done 
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362 

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363 
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)" 
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364 
apply (unfold mset_le_def, rule iffI, rule_tac x = "B  A" in exI) 
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365 
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2]) 
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366 
done 
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367 

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368 
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)" 
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369 
unfolding mset_le_def by auto 
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370 

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371 
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)" 
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372 
unfolding mset_le_def by auto 
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373 

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374 
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D" 
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375 
apply (unfold mset_le_def) 
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376 
apply auto 
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377 
apply (erule_tac x = a in allE)+ 
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378 
apply auto 
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379 
done 
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380 

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381 
lemma mset_le_add_left[simp]: "A \<le># A + B" 
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382 
unfolding mset_le_def by auto 
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383 

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384 
lemma mset_le_add_right[simp]: "B \<le># A + B" 
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385 
unfolding mset_le_def by auto 
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386 

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387 
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B" 
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388 
by (simp add: mset_le_def) 
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389 

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390 
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B  C = A + (B  C)" 
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391 
by (simp add: multiset_eq_conv_count_eq mset_le_def) 
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392 

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393 
lemma mset_le_multiset_union_diff_commute: 
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394 
assumes "B \<le># A" 
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395 
shows "A  B + C = A + C  B" 
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396 
proof  
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397 
from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" .. 
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398 
from this obtain D where "A = B + D" .. 
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399 
then show ?thesis 
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400 
apply simp 
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401 
apply (subst add_commute) 
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402 
apply (subst multiset_diff_union_assoc) 
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403 
apply simp 
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404 
apply (simp add: diff_cancel) 
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405 
apply (subst add_assoc) 
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406 
apply (subst add_commute [of "B" _]) 
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407 
apply (subst multiset_diff_union_assoc) 
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408 
apply simp 
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409 
apply (simp add: diff_cancel) 
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410 
done 
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411 
qed 
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412 

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413 
interpretation mset_order: order "op \<le>#" "op <#" 
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414 
proof qed (auto intro: order.intro mset_le_refl mset_le_antisym 
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415 
mset_le_trans simp: mset_less_def) 
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416 

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417 
interpretation mset_order_cancel_semigroup: 
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418 
pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#" 
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419 
proof qed (erule mset_le_mono_add [OF mset_le_refl]) 
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420 

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421 
interpretation mset_order_semigroup_cancel: 
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422 
pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#" 
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423 
proof qed simp 
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424 

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425 
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B" 
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426 
apply (clarsimp simp: mset_le_def mset_less_def) 
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427 
apply (erule_tac x=x in allE) 
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428 
apply auto 
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429 
done 
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changeset

430 

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431 
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B" 
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432 
apply (clarsimp simp: mset_le_def mset_less_def) 
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433 
apply (erule_tac x = x in allE) 
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434 
apply auto 
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435 
done 
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changeset

436 

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437 
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)" 
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438 
apply (rule conjI) 
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439 
apply (simp add: mset_lessD) 
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440 
apply (clarsimp simp: mset_le_def mset_less_def) 
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441 
apply safe 
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442 
apply (erule_tac x = a in allE) 
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443 
apply (auto split: split_if_asm) 
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444 
done 
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445 

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446 
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)" 
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447 
apply (rule conjI) 
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448 
apply (simp add: mset_leD) 
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449 
apply (force simp: mset_le_def mset_less_def split: split_if_asm) 
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450 
done 
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451 

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452 
lemma mset_less_of_empty[simp]: "A \<subset># {#} \<longleftrightarrow> False" 
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453 
by (auto simp add: mset_less_def mset_le_def multiset_eq_conv_count_eq) 
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454 

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455 
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}" 
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456 
by (auto simp: mset_le_def mset_less_def) 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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457 

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458 
lemma multi_psub_self[simp]: "A \<subset># A = False" 
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459 
by (auto simp: mset_le_def mset_less_def) 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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460 

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461 
lemma mset_less_add_bothsides: 
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462 
"T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S" 
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463 
by (auto simp: mset_le_def mset_less_def) 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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464 

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465 
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})" 
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466 
by (auto simp: mset_le_def mset_less_def) 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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changeset

467 

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468 
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B  {#c#} \<subset># B" 
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469 
by (auto simp: mset_le_def mset_less_def multiset_eq_conv_count_eq) 
10249  470 

471 

472 
subsubsection {* Set of elements *} 

473 

34943
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474 
definition set_of :: "'a multiset => 'a set" where 
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475 
"set_of M = {x. x :# M}" 
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476 

17161  477 
lemma set_of_empty [simp]: "set_of {#} = {}" 
26178  478 
by (simp add: set_of_def) 
10249  479 

17161  480 
lemma set_of_single [simp]: "set_of {#b#} = {b}" 
26178  481 
by (simp add: set_of_def) 
10249  482 

17161  483 
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N" 
26178  484 
by (auto simp add: set_of_def) 
10249  485 

17161  486 
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})" 
34943
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487 
by (auto simp add: set_of_def multiset_eq_conv_count_eq) 
10249  488 

17161  489 
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)" 
26178  490 
by (auto simp add: set_of_def) 
26016  491 

26033  492 
lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}" 
26178  493 
by (auto simp add: set_of_def) 
10249  494 

34943
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495 
lemma finite_set_of [iff]: "finite (set_of M)" 
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496 
using count [of M] by (simp add: multiset_def set_of_def) 
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497 

10249  498 

499 
subsubsection {* Size *} 

500 

34943
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501 
instantiation multiset :: (type) size 
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502 
begin 
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503 

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504 
definition size_def: 
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505 
"size M = setsum (count M) (set_of M)" 
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506 

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507 
instance .. 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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508 

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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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509 
end 
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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510 

28708
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511 
lemma size_empty [simp]: "size {#} = 0" 
26178  512 
by (simp add: size_def) 
10249  513 

28708
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514 
lemma size_single [simp]: "size {#b#} = 1" 
26178  515 
by (simp add: size_def) 
10249  516 

17161  517 
lemma setsum_count_Int: 
26178  518 
"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A" 
519 
apply (induct rule: finite_induct) 

520 
apply simp 

521 
apply (simp add: Int_insert_left set_of_def) 

522 
done 

10249  523 

28708
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524 
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N" 
26178  525 
apply (unfold size_def) 
526 
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)") 

527 
prefer 2 

528 
apply (rule ext, simp) 

529 
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int) 

530 
apply (subst Int_commute) 

531 
apply (simp (no_asm_simp) add: setsum_count_Int) 

532 
done 

10249  533 

17161  534 
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})" 
34943
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535 
by (auto simp add: size_def multiset_eq_conv_count_eq) 
26016  536 

537 
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)" 

26178  538 
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty) 
10249  539 

17161  540 
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M" 
26178  541 
apply (unfold size_def) 
542 
apply (drule setsum_SucD) 

543 
apply auto 

544 
done 

10249  545 

34943
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546 
lemma size_eq_Suc_imp_eq_union: 
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547 
assumes "size M = Suc n" 
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548 
shows "\<exists>a N. M = N + {#a#}" 
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549 
proof  
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550 
from assms obtain a where "a \<in># M" 
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551 
by (erule size_eq_Suc_imp_elem [THEN exE]) 
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552 
then have "M = M  {#a#} + {#a#}" by simp 
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553 
then show ?thesis by blast 
23611  554 
qed 
15869  555 

26016  556 

557 
subsection {* Induction and case splits *} 

10249  558 

559 
lemma setsum_decr: 

11701
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parents:
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diff
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560 
"finite F ==> (0::nat) < f a ==> 
15072  561 
setsum (f (a := f a  1)) F = (if a\<in>F then setsum f F  1 else setsum f F)" 
26178  562 
apply (induct rule: finite_induct) 
563 
apply auto 

564 
apply (drule_tac a = a in mk_disjoint_insert, auto) 

565 
done 

10249  566 

10313  567 
lemma rep_multiset_induct_aux: 
26178  568 
assumes 1: "P (\<lambda>a. (0::nat))" 
569 
and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))" 

570 
shows "\<forall>f. f \<in> multiset > setsum f {x. f x \<noteq> 0} = n > P f" 

571 
apply (unfold multiset_def) 

572 
apply (induct_tac n, simp, clarify) 

573 
apply (subgoal_tac "f = (\<lambda>a.0)") 

574 
apply simp 

575 
apply (rule 1) 

576 
apply (rule ext, force, clarify) 

577 
apply (frule setsum_SucD, clarify) 

578 
apply (rename_tac a) 

579 
apply (subgoal_tac "finite {x. (f (a := f a  1)) x > 0}") 

580 
prefer 2 

581 
apply (rule finite_subset) 

582 
prefer 2 

583 
apply assumption 

584 
apply simp 

585 
apply blast 

586 
apply (subgoal_tac "f = (f (a := f a  1))(a := (f (a := f a  1)) a + 1)") 

587 
prefer 2 

588 
apply (rule ext) 

589 
apply (simp (no_asm_simp)) 

590 
apply (erule ssubst, rule 2 [unfolded multiset_def], blast) 

591 
apply (erule allE, erule impE, erule_tac [2] mp, blast) 

592 
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def) 

593 
apply (subgoal_tac "{x. x \<noteq> a > f x \<noteq> 0} = {x. f x \<noteq> 0}") 

594 
prefer 2 

595 
apply blast 

596 
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0}  {a}") 

597 
prefer 2 

598 
apply blast 

599 
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong) 

600 
done 

10249  601 

10313  602 
theorem rep_multiset_induct: 
11464  603 
"f \<in> multiset ==> P (\<lambda>a. 0) ==> 
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11655
diff
changeset

604 
(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f" 
26178  605 
using rep_multiset_induct_aux by blast 
10249  606 

18258  607 
theorem multiset_induct [case_names empty add, induct type: multiset]: 
26178  608 
assumes empty: "P {#}" 
609 
and add: "!!M x. P M ==> P (M + {#x#})" 

610 
shows "P M" 

10249  611 
proof  
612 
note defns = union_def single_def Mempty_def 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

613 
note add' = add [unfolded defns, simplified] 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

614 
have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) = 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

615 
(\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset) 
10249  616 
show ?thesis 
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

617 
apply (rule count_inverse [THEN subst]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

618 
apply (rule count [THEN rep_multiset_induct]) 
18258  619 
apply (rule empty [unfolded defns]) 
15072  620 
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))") 
10249  621 
prefer 2 
622 
apply (simp add: expand_fun_eq) 

623 
apply (erule ssubst) 

17200  624 
apply (erule Abs_multiset_inverse [THEN subst]) 
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

625 
apply (drule add') 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

626 
apply (simp add: aux) 
10249  627 
done 
628 
qed 

629 

25610  630 
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}" 
26178  631 
by (induct M) auto 
25610  632 

633 
lemma multiset_cases [cases type, case_names empty add]: 

26178  634 
assumes em: "M = {#} \<Longrightarrow> P" 
635 
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P" 

636 
shows "P" 

25610  637 
proof (cases "M = {#}") 
26145  638 
assume "M = {#}" then show ?thesis using em by simp 
25610  639 
next 
640 
assume "M \<noteq> {#}" 

641 
then obtain M' m where "M = M' + {#m#}" 

642 
by (blast dest: multi_nonempty_split) 

26145  643 
then show ?thesis using add by simp 
25610  644 
qed 
645 

646 
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}" 

26178  647 
apply (cases M) 
648 
apply simp 

649 
apply (rule_tac x="M  {#x#}" in exI, simp) 

650 
done 

25610  651 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

652 
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B  {#c#} \<noteq> B" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

653 
by (cases "B = {#}") (auto dest: multi_member_split) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

654 

26033  655 
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}" 
26178  656 
apply (subst multiset_eq_conv_count_eq) 
657 
apply auto 

658 
done 

10249  659 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

660 
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

661 
proof (induct A arbitrary: B) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

662 
case (empty M) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

663 
then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

664 
then obtain M' x where "M = M' + {#x#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

665 
by (blast dest: multi_nonempty_split) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

666 
then show ?case by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

667 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

668 
case (add S x T) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

669 
have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

670 
have SxsubT: "S + {#x#} \<subset># T" by fact 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

671 
then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

672 
then obtain T' where T: "T = T' + {#x#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

673 
by (blast dest: multi_member_split) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

674 
then have "S \<subset># T'" using SxsubT 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

675 
by (blast intro: mset_less_add_bothsides) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

676 
then have "size S < size T'" using IH by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

677 
then show ?case using T by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

678 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

679 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

680 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

681 
subsubsection {* Strong induction and subset induction for multisets *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

682 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

683 
text {* Wellfoundedness of proper subset operator: *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

684 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

685 
text {* proper multiset subset *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

686 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

687 
definition 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

688 
mset_less_rel :: "('a multiset * 'a multiset) set" where 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

689 
"mset_less_rel = {(A,B). A \<subset># B}" 
10249  690 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

691 
lemma multiset_add_sub_el_shuffle: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

692 
assumes "c \<in># B" and "b \<noteq> c" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

693 
shows "B  {#c#} + {#b#} = B + {#b#}  {#c#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

694 
proof  
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

695 
from `c \<in># B` obtain A where B: "B = A + {#c#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

696 
by (blast dest: multi_member_split) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

697 
have "A + {#b#} = A + {#b#} + {#c#}  {#c#}" by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

698 
then have "A + {#b#} = A + {#c#} + {#b#}  {#c#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

699 
by (simp add: add_ac) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

700 
then show ?thesis using B by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

701 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

702 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

703 
lemma wf_mset_less_rel: "wf mset_less_rel" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

704 
apply (unfold mset_less_rel_def) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

705 
apply (rule wf_measure [THEN wf_subset, where f1=size]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

706 
apply (clarsimp simp: measure_def inv_image_def mset_less_size) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

707 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

708 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

709 
text {* The induction rules: *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

710 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

711 
lemma full_multiset_induct [case_names less]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

712 
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

713 
shows "P B" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

714 
apply (rule wf_mset_less_rel [THEN wf_induct]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

715 
apply (rule ih, auto simp: mset_less_rel_def) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

716 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

717 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

718 
lemma multi_subset_induct [consumes 2, case_names empty add]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

719 
assumes "F \<subseteq># A" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

720 
and empty: "P {#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

721 
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

722 
shows "P F" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

723 
proof  
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

724 
from `F \<subseteq># A` 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

725 
show ?thesis 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

726 
proof (induct F) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

727 
show "P {#}" by fact 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

728 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

729 
fix x F 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

730 
assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

731 
show "P (F + {#x#})" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

732 
proof (rule insert) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

733 
from i show "x \<in># A" by (auto dest: mset_le_insertD) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

734 
from i have "F \<subseteq># A" by (auto dest: mset_le_insertD) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

735 
with P show "P F" . 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

736 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

737 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

738 
qed 
26145  739 

17161  740 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

741 
subsection {* Alternative representations *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

742 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

743 
subsubsection {* Lists *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

744 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

745 
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

746 
"multiset_of [] = {#}"  
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

747 
"multiset_of (a # x) = multiset_of x + {# a #}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

748 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

749 
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

750 
by (induct x) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

751 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

752 
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

753 
by (induct x) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

754 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

755 
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

756 
by (induct x) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

757 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

758 
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

759 
by (induct xs) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

760 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

761 
lemma multiset_of_append [simp]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

762 
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

763 
by (induct xs arbitrary: ys) (auto simp: add_ac) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

764 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

765 
lemma surj_multiset_of: "surj multiset_of" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

766 
apply (unfold surj_def) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

767 
apply (rule allI) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

768 
apply (rule_tac M = y in multiset_induct) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

769 
apply auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

770 
apply (rule_tac x = "x # xa" in exI) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

771 
apply auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

772 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

773 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

774 
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

775 
by (induct x) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

776 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

777 
lemma distinct_count_atmost_1: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

778 
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

779 
apply (induct x, simp, rule iffI, simp_all) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

780 
apply (rule conjI) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

781 
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

782 
apply (erule_tac x = a in allE, simp, clarify) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

783 
apply (erule_tac x = aa in allE, simp) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

784 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

785 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

786 
lemma multiset_of_eq_setD: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

787 
"multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

788 
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

789 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

790 
lemma set_eq_iff_multiset_of_eq_distinct: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

791 
"distinct x \<Longrightarrow> distinct y \<Longrightarrow> 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

792 
(set x = set y) = (multiset_of x = multiset_of y)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

793 
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

794 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

795 
lemma set_eq_iff_multiset_of_remdups_eq: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

796 
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

797 
apply (rule iffI) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

798 
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

799 
apply (drule distinct_remdups [THEN distinct_remdups 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

800 
[THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

801 
apply simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

802 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

803 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

804 
lemma multiset_of_compl_union [simp]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

805 
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

806 
by (induct xs) (auto simp: add_ac) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

807 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

808 
lemma count_filter: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

809 
"count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

810 
by (induct xs) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

811 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

812 
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

813 
apply (induct ls arbitrary: i) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

814 
apply simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

815 
apply (case_tac i) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

816 
apply auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

817 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

818 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

819 
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs  {#a#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

820 
by (induct xs) (auto simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

821 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

822 
lemma multiset_of_eq_length: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

823 
assumes "multiset_of xs = multiset_of ys" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

824 
shows "length xs = length ys" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

825 
using assms 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

826 
proof (induct arbitrary: ys rule: length_induct) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

827 
case (1 xs ys) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

828 
show ?case 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

829 
proof (cases xs) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

830 
case Nil with "1.prems" show ?thesis by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

831 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

832 
case (Cons x xs') 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

833 
note xCons = Cons 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

834 
show ?thesis 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

835 
proof (cases ys) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

836 
case Nil 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

837 
with "1.prems" Cons show ?thesis by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

838 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

839 
case (Cons y ys') 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

840 
have x_in_ys: "x = y \<or> x \<in> set ys'" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

841 
proof (cases "x = y") 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

842 
case True then show ?thesis .. 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

843 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

844 
case False 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

845 
from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

846 
with False show ?thesis by (simp add: mem_set_multiset_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

847 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

848 
from "1.hyps" have IH: "length xs' < length xs \<longrightarrow> 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

849 
(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

850 
from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

851 
apply  
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

852 
apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

853 
apply fastsimp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

854 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

855 
with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

856 
from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

857 
with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

858 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

859 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

860 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

861 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

862 
text {* 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

863 
This lemma shows which properties suffice to show that a function 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

864 
@{text "f"} with @{text "f xs = ys"} behaves like sort. 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

865 
*} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

866 
lemma properties_for_sort: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

867 
"multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

868 
proof (induct xs arbitrary: ys) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

869 
case Nil then show ?case by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

870 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

871 
case (Cons x xs) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

872 
then have "x \<in> set ys" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

873 
by (auto simp add: mem_set_multiset_eq intro!: ccontr) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

874 
with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

875 
by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

876 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

877 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

878 
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

879 
apply (induct xs) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

880 
apply auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

881 
apply (rule mset_le_trans) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

882 
apply auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

883 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

884 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

885 
lemma multiset_of_update: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

886 
"i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls  {#ls ! i#} + {#v#}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

887 
proof (induct ls arbitrary: i) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

888 
case Nil then show ?case by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

889 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

890 
case (Cons x xs) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

891 
show ?case 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

892 
proof (cases i) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

893 
case 0 then show ?thesis by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

894 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

895 
case (Suc i') 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

896 
with Cons show ?thesis 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

897 
apply simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

898 
apply (subst add_assoc) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

899 
apply (subst add_commute [of "{#v#}" "{#x#}"]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

900 
apply (subst add_assoc [symmetric]) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

901 
apply simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

902 
apply (rule mset_le_multiset_union_diff_commute) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

903 
apply (simp add: mset_le_single nth_mem_multiset_of) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

904 
done 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

905 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

906 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

907 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

908 
lemma multiset_of_swap: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

909 
"i < length ls \<Longrightarrow> j < length ls \<Longrightarrow> 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

910 
multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

911 
by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

912 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

913 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

914 
subsubsection {* Association lists  including rudimentary code generation *} 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

915 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

916 
definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

917 
"count_of xs x = (case map_of xs x of None \<Rightarrow> 0  Some n \<Rightarrow> n)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

918 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

919 
lemma count_of_multiset: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

920 
"count_of xs \<in> multiset" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

921 
proof  
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

922 
let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat  Some (n\<Colon>nat) \<Rightarrow> n)}" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

923 
have "?A \<subseteq> dom (map_of xs)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

924 
proof 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

925 
fix x 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

926 
assume "x \<in> ?A" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

927 
then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat  Some (n\<Colon>nat) \<Rightarrow> n)" by simp 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

928 
then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

929 
then show "x \<in> dom (map_of xs)" by auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

930 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

931 
with finite_dom_map_of [of xs] have "finite ?A" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

932 
by (auto intro: finite_subset) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

933 
then show ?thesis 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

934 
by (simp add: count_of_def expand_fun_eq multiset_def) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

935 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

936 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

937 
lemma count_simps [simp]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

938 
"count_of [] = (\<lambda>_. 0)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

939 
"count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

940 
by (simp_all add: count_of_def expand_fun_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

941 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

942 
lemma count_of_empty: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

943 
"x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

944 
by (induct xs) (simp_all add: count_of_def) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

945 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

946 
lemma count_of_filter: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

947 
"count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

948 
by (induct xs) auto 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

949 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

950 
definition Bag :: "('a \<times> nat) list \<Rightarrow> 'a multiset" where 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

951 
"Bag xs = Abs_multiset (count_of xs)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

952 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

953 
code_datatype Bag 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

954 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

955 
lemma count_Bag [simp, code]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

956 
"count (Bag xs) = count_of xs" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

957 
by (simp add: Bag_def count_of_multiset Abs_multiset_inverse) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

958 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

959 
lemma Mempty_Bag [code]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

960 
"{#} = Bag []" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

961 
by (simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

962 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

963 
lemma single_Bag [code]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

964 
"{#x#} = Bag [(x, 1)]" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

965 
by (simp add: multiset_eq_conv_count_eq) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

966 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

967 
lemma MCollect_Bag [code]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

968 
"MCollect (Bag xs) P = Bag (filter (P \<circ> fst) xs)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

969 
by (simp add: multiset_eq_conv_count_eq count_of_filter) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

970 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

971 
lemma mset_less_eq_Bag [code]: 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

972 
"Bag xs \<subseteq># A \<longleftrightarrow> (\<forall>(x, n) \<in> set xs. count_of xs x \<le> count A x)" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

973 
(is "?lhs \<longleftrightarrow> ?rhs") 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

974 
proof 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

975 
assume ?lhs then show ?rhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

976 
by (auto simp add: mset_le_def count_Bag) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

977 
next 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

978 
assume ?rhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

979 
show ?lhs 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

980 
proof (rule mset_less_eqI) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

981 
fix x 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

982 
from `?rhs` have "count_of xs x \<le> count A x" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

983 
by (cases "x \<in> fst ` set xs") (auto simp add: count_of_empty) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

984 
then show "count (Bag xs) x \<le> count A x" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

985 
by (simp add: mset_le_def count_Bag) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

986 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

987 
qed 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

988 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

989 
instantiation multiset :: (eq) eq 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

990 
begin 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

991 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

992 
definition 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

993 
"HOL.eq A B \<longleftrightarrow> A \<subseteq># B \<and> B \<subseteq># A" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

994 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

995 
instance proof 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

996 
qed (simp add: eq_multiset_def mset_order.eq_iff) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

997 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

998 
end 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

999 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1000 
definition (in term_syntax) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1001 
bagify :: "('a\<Colon>typerep \<times> nat) list \<times> (unit \<Rightarrow> Code_Evaluation.term) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1002 
\<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1003 
[code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1004 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1005 
notation fcomp (infixl "o>" 60) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1006 
notation scomp (infixl "o\<rightarrow>" 60) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1007 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1008 
instantiation multiset :: (random) random 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1009 
begin 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1010 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1011 
definition 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1012 
"Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>xs. Pair (bagify xs))" 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1013 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1014 
instance .. 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1015 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1016 
end 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1017 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1018 
no_notation fcomp (infixl "o>" 60) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1019 
no_notation scomp (infixl "o\<rightarrow>" 60) 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1020 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1021 
hide (open) const bagify 
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1022 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1023 

e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1024 
subsection {* The multiset order *} 
10249  1025 

1026 
subsubsection {* Wellfoundedness *} 

1027 

28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset

1028 
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where 
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset

1029 
[code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> 
23751  1030 
(\<forall>b. b :# K > (b, a) \<in> r)}" 
10249  1031 

28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset

1032 
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where 
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1033 
[code del]: "mult r = (mult1 r)\<^sup>+" 
10249  1034 

23751  1035 
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r" 
26178  1036 
by (simp add: mult1_def) 
10249  1037 

23751  1038 
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==> 
1039 
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or> 

1040 
(\<exists>K. (\<forall>b. b :# K > (b, a) \<in> r) \<and> N = M0 + K)" 

19582  1041 
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2") 
10249  1042 
proof (unfold mult1_def) 
23751  1043 
let ?r = "\<lambda>K a. \<forall>b. b :# K > (b, a) \<in> r" 
11464  1044 
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a" 
23751  1045 
let ?case1 = "?case1 {(N, M). ?R N M}" 
10249  1046 

23751  1047 
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}" 
18258  1048 
then have "\<exists>a' M0' K. 
11464  1049 
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp 
18258  1050 
then show "?case1 \<or> ?case2" 
10249  1051 
proof (elim exE conjE) 
1052 
fix a' M0' K 

1053 
assume N: "N = M0' + K" and r: "?r K a'" 

1054 
assume "M0 + {#a#} = M0' + {#a'#}" 

18258  1055 
then have "M0 = M0' \<and> a = a' \<or> 
11464  1056 
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})" 
10249  1057 
by (simp only: add_eq_conv_ex) 
18258  1058 
then show ?thesis 
10249  1059 
proof (elim disjE conjE exE) 
1060 
assume "M0 = M0'" "a = a'" 

11464  1061 
with N r have "?r K a \<and> N = M0 + K" by simp 
18258  1062 
then have ?case2 .. then show ?thesis .. 
10249  1063 
next 
1064 
fix K' 

1065 
assume "M0' = K' + {#a#}" 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1066 
with N have n: "N = K' + K + {#a#}" by (simp add: add_ac) 
10249  1067 

1068 
assume "M0 = K' + {#a'#}" 

1069 
with r have "?R (K' + K) M0" by blast 

18258  1070 
with n have ?case1 by simp then show ?thesis .. 
10249  1071 
qed 
1072 
qed 

1073 
qed 

1074 

23751  1075 
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)" 
10249  1076 
proof 
1077 
let ?R = "mult1 r" 

1078 
let ?W = "acc ?R" 

1079 
{ 

1080 
fix M M0 a 

23751  1081 
assume M0: "M0 \<in> ?W" 
1082 
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" 

1083 
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R > M + {#a#} \<in> ?W" 

1084 
have "M0 + {#a#} \<in> ?W" 

1085 
proof (rule accI [of "M0 + {#a#}"]) 

10249  1086 
fix N 
23751  1087 
assume "(N, M0 + {#a#}) \<in> ?R" 
1088 
then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or> 

1089 
(\<exists>K. (\<forall>b. b :# K > (b, a) \<in> r) \<and> N = M0 + K))" 

10249  1090 
by (rule less_add) 
23751  1091 
then show "N \<in> ?W" 
10249  1092 
proof (elim exE disjE conjE) 
23751  1093 
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}" 
1094 
from acc_hyp have "(M, M0) \<in> ?R > M + {#a#} \<in> ?W" .. 

1095 
from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" .. 

1096 
then show "N \<in> ?W" by (simp only: N) 

10249  1097 
next 
1098 
fix K 

1099 
assume N: "N = M0 + K" 

23751  1100 
assume "\<forall>b. b :# K > (b, a) \<in> r" 
1101 
then have "M0 + K \<in> ?W" 

10249  1102 
proof (induct K) 
18730  1103 
case empty 
23751  1104 
from M0 show "M0 + {#} \<in> ?W" by simp 
18730  1105 
next 
1106 
case (add K x) 

23751  1107 
from add.prems have "(x, a) \<in> r" by simp 
1108 
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast 

1109 
moreover from add have "M0 + K \<in> ?W" by simp 

1110 
ultimately have "(M0 + K) + {#x#} \<in> ?W" .. 

34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset

1111 
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc) 
10249  1112 
qed 
23751  1113 
then show "N \<in> ?W" by (simp only: N) 
10249  1114 
qed 
1115 
qed 

1116 
} note tedious_reasoning = this 

1117 

23751  1118 
assume wf: "wf r" 
10249  1119 
fix M 
23751  1120 
show "M \<in> ?W" 
10249  1121 
proof (induct M) 
23751  1122 
show "{#} \<in> ?W" 
10249  1123 
proof (rule accI) 
23751  1124 
fix b assume "(b, {#}) \<in> ?R" 
1125 
with not_less_empty show "b \<in> ?W" by contradiction 

10249  1126 
qed 
1127 

23751  1128 
fix M a assume "M \<in> ?W" 
1129 
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" 

10249  1130 
proof induct 
1131 
fix a 

23751  1132 
assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" 
1133 
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" 

10249  1134 
proof 
23751  1135 
fix M assume "M \<in> ?W" 
1136 
then show "M + {#a#} \<in> ?W" 
