src/HOL/Library/More_Set.thy
 author haftmann Fri Aug 27 19:34:23 2010 +0200 (2010-08-27 ago) changeset 38857 97775f3e8722 parent 37595 9591362629e3 child 39198 f967a16dfcdd permissions -rw-r--r--
renamed class/constant eq to equal; tuned some instantiations
```     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 More_Set
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```     7 imports Main More_List
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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"
```
```    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 = Finite_Set.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> List.null xs"
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```    41   by (simp add: is_empty_def null_def)
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```    42
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```    43 lemma empty_set:
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```    44   "{} = set []"
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```    45   by simp
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```    46
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```    47 lemma insert_set_compl:
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```    48   "insert x (- set xs) = - set (removeAll x xs)"
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```    49   by auto
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```    50
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```    51 lemma remove_set_compl:
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```    52   "remove x (- set xs) = - set (List.insert x xs)"
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```    53   by (auto simp del: mem_def simp add: remove_def List.insert_def)
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```    54
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```    55 lemma image_set:
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```    56   "image f (set xs) = set (map f xs)"
```
```    57   by simp
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```    58
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```    59 lemma project_set:
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```    60   "project P (set xs) = set (filter P xs)"
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```    61   by (auto simp add: project_def)
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```    62
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```    63
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```    64 subsection {* Functorial set operations *}
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```    65
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```    66 lemma union_set:
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```    67   "set xs \<union> A = fold Set.insert xs A"
```
```    68 proof -
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```    69   interpret fun_left_comm_idem Set.insert
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```    70     by (fact fun_left_comm_idem_insert)
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```    71   show ?thesis by (simp add: union_fold_insert fold_set)
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```    72 qed
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```    73
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```    74 lemma union_set_foldr:
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```    75   "set xs \<union> A = foldr Set.insert xs A"
```
```    76 proof -
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```    77   have "\<And>x y :: 'a. insert y \<circ> insert x = insert x \<circ> insert y"
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```    78     by (auto intro: ext)
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```    79   then show ?thesis by (simp add: union_set foldr_fold)
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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 = fold remove xs A"
```
```    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 lemma minus_set_foldr:
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```    92   "A - set xs = foldr remove xs A"
```
```    93 proof -
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```    94   have "\<And>x y :: 'a. remove y \<circ> remove x = remove x \<circ> remove y"
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```    95     by (auto simp add: remove_def intro: ext)
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```    96   then show ?thesis by (simp add: minus_set foldr_fold)
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```    97 qed
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```    98
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```    99
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```   100 subsection {* Derived set operations *}
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```   101
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```   102 lemma member:
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```   103   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"
```
```   104   by simp
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```   105
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```   106 lemma subset_eq:
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```   107   "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
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```   108   by (fact subset_eq)
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```   109
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```   110 lemma subset:
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```   111   "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
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```   112   by (fact less_le_not_le)
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```   113
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```   114 lemma set_eq:
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```   115   "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
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```   116   by (fact eq_iff)
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```   117
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```   118 lemma inter:
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```   119   "A \<inter> B = project (\<lambda>x. x \<in> A) B"
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```   120   by (auto simp add: project_def)
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```   121
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```   122
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```   123 subsection {* Various lemmas *}
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```   124
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```   125 lemma not_set_compl:
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```   126   "Not \<circ> set xs = - set xs"
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```   127   by (simp add: fun_Compl_def bool_Compl_def comp_def expand_fun_eq)
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```   128
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```   129 end
```