src/HOL/Library/List_Set.thy
 author wenzelm Thu Feb 11 23:00:22 2010 +0100 (2010-02-11) changeset 35115 446c5063e4fd parent 34977 27ceb64d41ea child 37023 efc202e1677e permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
```     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 set functions *}
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```    11
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```    12 definition is_empty :: "'a set \<Rightarrow> bool" where
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```    13   "is_empty A \<longleftrightarrow> A = {}"
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```    14
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```    15 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
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```    16   "remove x A = A - {x}"
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```    17
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```    18 lemma fun_left_comm_idem_remove:
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```    19   "fun_left_comm_idem remove"
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```    20 proof -
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```    21   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)
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```    22   show ?thesis by (simp only: fun_left_comm_idem_remove rem)
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```    23 qed
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```    24
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```    25 lemma minus_fold_remove:
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```    26   assumes "finite A"
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```    27   shows "B - A = fold remove B A"
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```    28 proof -
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```    29   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)
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```    30   show ?thesis by (simp only: rem assms minus_fold_remove)
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```    31 qed
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```    32
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```    33 definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
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```    34   "project P A = {a\<in>A. P a}"
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```    35
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```    36
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```    37 subsection {* Basic set operations *}
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```    38
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```    39 lemma is_empty_set:
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```    40   "is_empty (set xs) \<longleftrightarrow> null xs"
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```    41   by (simp add: is_empty_def null_empty)
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```    42
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```    43 lemma ball_set:
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```    44   "(\<forall>x\<in>set xs. P x) \<longleftrightarrow> list_all P xs"
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```    45   by (rule list_ball_code)
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```    46
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```    47 lemma bex_set:
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```    48   "(\<exists>x\<in>set xs. P x) \<longleftrightarrow> list_ex P xs"
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```    49   by (rule list_bex_code)
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```    50
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```    51 lemma empty_set:
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```    52   "{} = set []"
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```    53   by simp
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```    54
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```    55 lemma insert_set_compl:
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```    56   "insert x (- set xs) = - set (removeAll x xs)"
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```    57   by auto
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```    58
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```    59 lemma remove_set_compl:
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```    60   "remove x (- set xs) = - set (List.insert x xs)"
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```    61   by (auto simp del: mem_def simp add: remove_def List.insert_def)
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```    62
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```    63 lemma image_set:
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```    64   "image f (set xs) = set (map f xs)"
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```    65   by simp
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```    66
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```    67 lemma project_set:
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```    68   "project P (set xs) = set (filter P xs)"
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```    69   by (auto simp add: project_def)
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```    70
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```    71
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```    72 subsection {* Functorial set operations *}
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```    73
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```    74 lemma union_set:
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```    75   "set xs \<union> A = foldl (\<lambda>A x. Set.insert x A) A xs"
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```    76 proof -
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```    77   interpret fun_left_comm_idem Set.insert
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```    78     by (fact fun_left_comm_idem_insert)
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```    79   show ?thesis by (simp add: union_fold_insert fold_set)
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```    80 qed
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```    81
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```    82 lemma minus_set:
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```    83   "A - set xs = foldl (\<lambda>A x. remove x A) A xs"
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```    84 proof -
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```    85   interpret fun_left_comm_idem remove
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```    86     by (fact fun_left_comm_idem_remove)
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```    87   show ?thesis
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```    88     by (simp add: minus_fold_remove [of _ A] fold_set)
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```    89 qed
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```    90
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```    91
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```    92 subsection {* Derived set operations *}
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```    93
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```    94 lemma member:
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```    95   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"
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```    96   by simp
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```    97
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```    98 lemma subset_eq:
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```    99   "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
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```   100   by (fact subset_eq)
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```   101
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```   102 lemma subset:
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```   103   "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
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```   104   by (fact less_le_not_le)
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```   105
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```   106 lemma set_eq:
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```   107   "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
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```   108   by (fact eq_iff)
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```   109
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```   110 lemma inter:
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```   111   "A \<inter> B = project (\<lambda>x. x \<in> A) B"
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```   112   by (auto simp add: project_def)
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```   113
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`   114 end`