src/HOL/Library/List_Set.thy
 changeset 31807 039893a9a77d child 31846 89c37daebfdd
equal inserted replaced
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`     1 `
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`     2 (* Author: Florian Haftmann, TU Muenchen *)`
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`     3 `
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`     4 header {* Relating (finite) sets and lists *}`
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`     5 `
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`     6 theory List_Set`
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`     7 imports Main`
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`     8 begin`
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`     9 `
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`    10 subsection {* Various additional list functions *}`
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`    11 `
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`    12 definition insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where`
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`    13   "insert x xs = (if x \<in> set xs then xs else x # xs)"`
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`    14 `
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`    15 definition remove_all :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where`
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`    16   "remove_all x xs = filter (Not o op = x) xs"`
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`    17 `
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`    18 `
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`    19 subsection {* Various additional set functions *}`
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`    20 `
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`    21 definition is_empty :: "'a set \<Rightarrow> bool" where`
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`    22   "is_empty A \<longleftrightarrow> A = {}"`
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`    23 `
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`    24 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where`
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`    25   "remove x A = A - {x}"`
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`    26 `
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`    27 lemma fun_left_comm_idem_remove:`
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`    28   "fun_left_comm_idem remove"`
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`    29 proof -`
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`    30   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)`
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`    31   show ?thesis by (simp only: fun_left_comm_idem_remove rem)`
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`    32 qed`
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`    33 `
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`    34 lemma minus_fold_remove:`
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`    35   assumes "finite A"`
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`    36   shows "B - A = fold remove B A"`
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`    37 proof -`
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`    38   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)`
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`    39   show ?thesis by (simp only: rem assms minus_fold_remove)`
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`    40 qed`
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`    41 `
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`    42 definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where`
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`    43   "project P A = {a\<in>A. P a}"`
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`    44 `
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`    45 `
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`    46 subsection {* Basic set operations *}`
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`    47 `
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`    48 lemma is_empty_set:`
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`    49   "is_empty (set xs) \<longleftrightarrow> null xs"`
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`    50   by (simp add: is_empty_def null_empty)`
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`    51 `
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`    52 lemma ball_set:`
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`    53   "(\<forall>x\<in>set xs. P x) \<longleftrightarrow> list_all P xs"`
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`    54   by (rule list_ball_code)`
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`    55 `
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`    56 lemma bex_set:`
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`    57   "(\<exists>x\<in>set xs. P x) \<longleftrightarrow> list_ex P xs"`
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`    58   by (rule list_bex_code)`
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`    59 `
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`    60 lemma empty_set:`
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`    61   "{} = set []"`
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`    62   by simp`
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`    63 `
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`    64 lemma insert_set:`
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`    65   "Set.insert x (set xs) = set (insert x xs)"`
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`    66   by (auto simp add: insert_def)`
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`    67 `
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`    68 lemma remove_set:`
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`    69   "remove x (set xs) = set (remove_all x xs)"`
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`    70   by (auto simp add: remove_def remove_all_def)`
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`    71 `
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`    72 lemma image_set:`
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`    73   "image f (set xs) = set (remdups (map f xs))"`
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`    74   by simp`
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`    75 `
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`    76 lemma project_set:`
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`    77   "project P (set xs) = set (filter P xs)"`
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`    78   by (auto simp add: project_def)`
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`    79 `
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`    80 `
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`    81 subsection {* Functorial set operations *}`
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`    82 `
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`    83 lemma union_set:`
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`    84   "set xs \<union> A = foldl (\<lambda>A x. Set.insert x A) A xs"`
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`    85 proof -`
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`    86   interpret fun_left_comm_idem Set.insert`
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`    87     by (fact fun_left_comm_idem_insert)`
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`    88   show ?thesis by (simp add: union_fold_insert fold_set)`
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`    89 qed`
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`    90 `
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`    91 lemma minus_set:`
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`    92   "A - set xs = foldl (\<lambda>A x. remove x A) A xs"`
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`    93 proof -`
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`    94   interpret fun_left_comm_idem remove`
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`    95     by (fact fun_left_comm_idem_remove)`
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`    96   show ?thesis`
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`    97     by (simp add: minus_fold_remove [of _ A] fold_set)`
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`    98 qed`
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`    99 `
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`   100 lemma Inter_set:`
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`   101   "Inter (set (A # As)) = foldl (op \<inter>) A As"`
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`   102 proof -`
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`   103   have "finite (set (A # As))" by simp`
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`   104   moreover have "fold (op \<inter>) UNIV (set (A # As)) = foldl (\<lambda>y x. x \<inter> y) UNIV (A # As)"`
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`   105     by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)`
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`   106   ultimately have "Inter (set (A # As)) = foldl (op \<inter>) UNIV (A # As)"`
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`   107     by (simp only: Inter_fold_inter Int_commute)`
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`   108   then show ?thesis by simp`
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`   109 qed`
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`   110 `
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`   111 lemma Union_set:`
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`   112   "Union (set As) = foldl (op \<union>) {} As"`
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`   113 proof -`
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`   114   have "fold (op \<union>) {} (set As) = foldl (\<lambda>y x. x \<union> y) {} As"`
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`   115     by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)`
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`   116   then show ?thesis`
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`   117     by (simp only: Union_fold_union finite_set Un_commute)`
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`   118 qed`
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`   119 `
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`   120 lemma INTER_set:`
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`   121   "INTER (set (A # As)) f = foldl (\<lambda>B A. f A \<inter> B) (f A) As"`
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`   122 proof -`
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`   123   have "finite (set (A # As))" by simp`
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`   124   moreover have "fold (\<lambda>A. op \<inter> (f A)) UNIV (set (A # As)) = foldl (\<lambda>B A. f A \<inter> B) UNIV (A # As)"`
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`   125     by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)`
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`   126   ultimately have "INTER (set (A # As)) f = foldl (\<lambda>B A. f A \<inter> B) UNIV (A # As)"`
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`   127     by (simp only: INTER_fold_inter) `
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`   128   then show ?thesis by simp`
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`   129 qed`
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`   130 `
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`   131 lemma UNION_set:`
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`   132   "UNION (set As) f = foldl (\<lambda>B A. f A \<union> B) {} As"`
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`   133 proof -`
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`   134   have "fold (\<lambda>A. op \<union> (f A)) {} (set As) = foldl (\<lambda>B A. f A \<union> B) {} As"`
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`   135     by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)`
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`   136   then show ?thesis`
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`   137     by (simp only: UNION_fold_union finite_set)`
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`   138 qed`
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`   139 `
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`   140 `
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`   141 subsection {* Derived set operations *}`
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`   142 `
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`   143 lemma member:`
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`   144   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"`
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`   145   by simp`
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`   146 `
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`   147 lemma subset_eq:`
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`   148   "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"`
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`   149   by (fact subset_eq)`
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`   150 `
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`   151 lemma subset:`
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`   152   "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"`
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`   153   by (fact less_le_not_le)`
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`   154 `
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`   155 lemma set_eq:`
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`   156   "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"`
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`   157   by (fact eq_iff)`
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`   158 `
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`   159 lemma inter:`
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`   160   "A \<inter> B = project (\<lambda>x. x \<in> A) B"`
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`   161   by (auto simp add: project_def)`
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`   162 `
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`   163 end`