src/HOL/Library/More_Set.thy
 author kuncar Fri Dec 09 18:07:04 2011 +0100 (2011-12-09) changeset 45802 b16f976db515 parent 45012 060f76635bfe child 45974 2b043ed911ac permissions -rw-r--r--
Quotient_Info stores only relation maps
```     1
```
```     2 (* Author: Florian Haftmann, TU Muenchen *)
```
```     3
```
```     4 header {* Relating (finite) sets and lists *}
```
```     5
```
```     6 theory More_Set
```
```     7 imports Main More_List
```
```     8 begin
```
```     9
```
```    10 subsection {* Various additional set functions *}
```
```    11
```
```    12 definition is_empty :: "'a set \<Rightarrow> bool" where
```
```    13   "is_empty A \<longleftrightarrow> A = {}"
```
```    14
```
```    15 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
```
```    16   "remove x A = A - {x}"
```
```    17
```
```    18 lemma comp_fun_idem_remove:
```
```    19   "comp_fun_idem remove"
```
```    20 proof -
```
```    21   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: fun_eq_iff remove_def)
```
```    22   show ?thesis by (simp only: comp_fun_idem_remove rem)
```
```    23 qed
```
```    24
```
```    25 lemma minus_fold_remove:
```
```    26   assumes "finite A"
```
```    27   shows "B - A = Finite_Set.fold remove B A"
```
```    28 proof -
```
```    29   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: fun_eq_iff remove_def)
```
```    30   show ?thesis by (simp only: rem assms minus_fold_remove)
```
```    31 qed
```
```    32
```
```    33 definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
```
```    34   "project P A = {a \<in> A. P a}"
```
```    35
```
```    36 lemma bounded_Collect_code [code_unfold_post]:
```
```    37   "{x \<in> A. P x} = project P A"
```
```    38   by (simp add: project_def)
```
```    39
```
```    40 definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
```
```    41   "product A B = Sigma A (\<lambda>_. B)"
```
```    42
```
```    43 hide_const (open) product
```
```    44
```
```    45 lemma [code_unfold_post]:
```
```    46   "Sigma A (\<lambda>_. B) = More_Set.product A B"
```
```    47   by (simp add: product_def)
```
```    48
```
```    49
```
```    50 subsection {* Basic set operations *}
```
```    51
```
```    52 lemma is_empty_set:
```
```    53   "is_empty (set xs) \<longleftrightarrow> List.null xs"
```
```    54   by (simp add: is_empty_def null_def)
```
```    55
```
```    56 lemma empty_set:
```
```    57   "{} = set []"
```
```    58   by simp
```
```    59
```
```    60 lemma insert_set_compl:
```
```    61   "insert x (- set xs) = - set (removeAll x xs)"
```
```    62   by auto
```
```    63
```
```    64 lemma remove_set_compl:
```
```    65   "remove x (- set xs) = - set (List.insert x xs)"
```
```    66   by (auto simp add: remove_def List.insert_def)
```
```    67
```
```    68 lemma image_set:
```
```    69   "image f (set xs) = set (map f xs)"
```
```    70   by simp
```
```    71
```
```    72 lemma project_set:
```
```    73   "project P (set xs) = set (filter P xs)"
```
```    74   by (auto simp add: project_def)
```
```    75
```
```    76
```
```    77 subsection {* Functorial set operations *}
```
```    78
```
```    79 lemma union_set:
```
```    80   "set xs \<union> A = fold Set.insert xs A"
```
```    81 proof -
```
```    82   interpret comp_fun_idem Set.insert
```
```    83     by (fact comp_fun_idem_insert)
```
```    84   show ?thesis by (simp add: union_fold_insert fold_set)
```
```    85 qed
```
```    86
```
```    87 lemma union_set_foldr:
```
```    88   "set xs \<union> A = foldr Set.insert xs A"
```
```    89 proof -
```
```    90   have "\<And>x y :: 'a. insert y \<circ> insert x = insert x \<circ> insert y"
```
```    91     by auto
```
```    92   then show ?thesis by (simp add: union_set foldr_fold)
```
```    93 qed
```
```    94
```
```    95 lemma minus_set:
```
```    96   "A - set xs = fold remove xs A"
```
```    97 proof -
```
```    98   interpret comp_fun_idem remove
```
```    99     by (fact comp_fun_idem_remove)
```
```   100   show ?thesis
```
```   101     by (simp add: minus_fold_remove [of _ A] fold_set)
```
```   102 qed
```
```   103
```
```   104 lemma minus_set_foldr:
```
```   105   "A - set xs = foldr remove xs A"
```
```   106 proof -
```
```   107   have "\<And>x y :: 'a. remove y \<circ> remove x = remove x \<circ> remove y"
```
```   108     by (auto simp add: remove_def)
```
```   109   then show ?thesis by (simp add: minus_set foldr_fold)
```
```   110 qed
```
```   111
```
```   112
```
```   113 subsection {* Derived set operations *}
```
```   114
```
```   115 lemma member:
```
```   116   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"
```
```   117   by simp
```
```   118
```
```   119 lemma subset_eq:
```
```   120   "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
```
```   121   by (fact subset_eq)
```
```   122
```
```   123 lemma subset:
```
```   124   "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
```
```   125   by (fact less_le_not_le)
```
```   126
```
```   127 lemma set_eq:
```
```   128   "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
```
```   129   by (fact eq_iff)
```
```   130
```
```   131 lemma inter:
```
```   132   "A \<inter> B = project (\<lambda>x. x \<in> A) B"
```
```   133   by (auto simp add: project_def)
```
```   134
```
```   135
```
```   136 subsection {* Theorems on relations *}
```
```   137
```
```   138 lemma product_code:
```
```   139   "More_Set.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
```
```   140   by (auto simp add: product_def)
```
```   141
```
```   142 lemma Id_on_set:
```
```   143   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
```
```   144   by (auto simp add: Id_on_def)
```
```   145
```
```   146 lemma set_rel_comp:
```
```   147   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
```
```   148   by (auto simp add: Bex_def)
```
```   149
```
```   150 lemma wf_set:
```
```   151   "wf (set xs) = acyclic (set xs)"
```
```   152   by (simp add: wf_iff_acyclic_if_finite)
```
```   153
```
```   154 end
```