src/HOL/Library/List_Set.thy
 author haftmann Thu Jun 25 17:07:18 2009 +0200 (2009-06-25) changeset 31807 039893a9a77d child 31846 89c37daebfdd permissions -rw-r--r--
added List_Set and Code_Set theories
```     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}"
```
```    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)
```
```    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)
```
```    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)"
```
```    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"
```
```    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"
```
```   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)"
```
```   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"
```
```   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)
```
```   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"
```
```   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)"
```
```   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"
```
```   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"
```
```   135     by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
```
```   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`