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;
 haftmann@31807 ` 1` haftmann@31807 ` 2` ```(* Author: Florian Haftmann, TU Muenchen *) ``` haftmann@31807 ` 3` haftmann@31807 ` 4` ```header {* Relating (finite) sets and lists *} ``` haftmann@31807 ` 5` haftmann@31807 ` 6` ```theory List_Set ``` haftmann@31807 ` 7` ```imports Main ``` haftmann@31807 ` 8` ```begin ``` haftmann@31807 ` 9` haftmann@31807 ` 10` ```subsection {* Various additional set functions *} ``` haftmann@31807 ` 11` haftmann@31807 ` 12` ```definition is_empty :: "'a set \ bool" where ``` haftmann@31807 ` 13` ``` "is_empty A \ A = {}" ``` haftmann@31807 ` 14` haftmann@31807 ` 15` ```definition remove :: "'a \ 'a set \ 'a set" where ``` haftmann@31807 ` 16` ``` "remove x A = A - {x}" ``` haftmann@31807 ` 17` haftmann@31807 ` 18` ```lemma fun_left_comm_idem_remove: ``` haftmann@31807 ` 19` ``` "fun_left_comm_idem remove" ``` haftmann@31807 ` 20` ```proof - ``` haftmann@31807 ` 21` ``` have rem: "remove = (\x A. A - {x})" by (simp add: expand_fun_eq remove_def) ``` haftmann@31807 ` 22` ``` show ?thesis by (simp only: fun_left_comm_idem_remove rem) ``` haftmann@31807 ` 23` ```qed ``` haftmann@31807 ` 24` haftmann@31807 ` 25` ```lemma minus_fold_remove: ``` haftmann@31807 ` 26` ``` assumes "finite A" ``` haftmann@31807 ` 27` ``` shows "B - A = fold remove B A" ``` haftmann@31807 ` 28` ```proof - ``` haftmann@31807 ` 29` ``` have rem: "remove = (\x A. A - {x})" by (simp add: expand_fun_eq remove_def) ``` haftmann@31807 ` 30` ``` show ?thesis by (simp only: rem assms minus_fold_remove) ``` haftmann@31807 ` 31` ```qed ``` haftmann@31807 ` 32` haftmann@31807 ` 33` ```definition project :: "('a \ bool) \ 'a set \ 'a set" where ``` haftmann@31807 ` 34` ``` "project P A = {a\A. P a}" ``` haftmann@31807 ` 35` haftmann@31807 ` 36` haftmann@31807 ` 37` ```subsection {* Basic set operations *} ``` haftmann@31807 ` 38` haftmann@31807 ` 39` ```lemma is_empty_set: ``` haftmann@31807 ` 40` ``` "is_empty (set xs) \ null xs" ``` haftmann@31807 ` 41` ``` by (simp add: is_empty_def null_empty) ``` haftmann@31807 ` 42` haftmann@31807 ` 43` ```lemma ball_set: ``` haftmann@31807 ` 44` ``` "(\x\set xs. P x) \ list_all P xs" ``` haftmann@31807 ` 45` ``` by (rule list_ball_code) ``` haftmann@31807 ` 46` haftmann@31807 ` 47` ```lemma bex_set: ``` haftmann@31807 ` 48` ``` "(\x\set xs. P x) \ list_ex P xs" ``` haftmann@31807 ` 49` ``` by (rule list_bex_code) ``` haftmann@31807 ` 50` haftmann@31807 ` 51` ```lemma empty_set: ``` haftmann@31807 ` 52` ``` "{} = set []" ``` haftmann@31807 ` 53` ``` by simp ``` haftmann@31807 ` 54` haftmann@32880 ` 55` ```lemma insert_set_compl: ``` haftmann@34977 ` 56` ``` "insert x (- set xs) = - set (removeAll x xs)" ``` haftmann@34977 ` 57` ``` by auto ``` haftmann@31807 ` 58` haftmann@32880 ` 59` ```lemma remove_set_compl: ``` haftmann@34977 ` 60` ``` "remove x (- set xs) = - set (List.insert x xs)" ``` haftmann@34977 ` 61` ``` by (auto simp del: mem_def simp add: remove_def List.insert_def) ``` haftmann@32880 ` 62` haftmann@31807 ` 63` ```lemma image_set: ``` haftmann@31846 ` 64` ``` "image f (set xs) = set (map f xs)" ``` haftmann@31807 ` 65` ``` by simp ``` haftmann@31807 ` 66` haftmann@31807 ` 67` ```lemma project_set: ``` haftmann@31807 ` 68` ``` "project P (set xs) = set (filter P xs)" ``` haftmann@31807 ` 69` ``` by (auto simp add: project_def) ``` haftmann@31807 ` 70` haftmann@31807 ` 71` haftmann@31807 ` 72` ```subsection {* Functorial set operations *} ``` haftmann@31807 ` 73` haftmann@31807 ` 74` ```lemma union_set: ``` haftmann@31807 ` 75` ``` "set xs \ A = foldl (\A x. Set.insert x A) A xs" ``` haftmann@31807 ` 76` ```proof - ``` haftmann@31807 ` 77` ``` interpret fun_left_comm_idem Set.insert ``` haftmann@31807 ` 78` ``` by (fact fun_left_comm_idem_insert) ``` haftmann@31807 ` 79` ``` show ?thesis by (simp add: union_fold_insert fold_set) ``` haftmann@31807 ` 80` ```qed ``` haftmann@31807 ` 81` haftmann@31807 ` 82` ```lemma minus_set: ``` haftmann@31807 ` 83` ``` "A - set xs = foldl (\A x. remove x A) A xs" ``` haftmann@31807 ` 84` ```proof - ``` haftmann@31807 ` 85` ``` interpret fun_left_comm_idem remove ``` haftmann@31807 ` 86` ``` by (fact fun_left_comm_idem_remove) ``` haftmann@31807 ` 87` ``` show ?thesis ``` haftmann@31807 ` 88` ``` by (simp add: minus_fold_remove [of _ A] fold_set) ``` haftmann@31807 ` 89` ```qed ``` haftmann@31807 ` 90` haftmann@31807 ` 91` haftmann@31807 ` 92` ```subsection {* Derived set operations *} ``` haftmann@31807 ` 93` haftmann@31807 ` 94` ```lemma member: ``` haftmann@31807 ` 95` ``` "a \ A \ (\x\A. a = x)" ``` haftmann@31807 ` 96` ``` by simp ``` haftmann@31807 ` 97` haftmann@31807 ` 98` ```lemma subset_eq: ``` haftmann@31807 ` 99` ``` "A \ B \ (\x\A. x \ B)" ``` haftmann@31807 ` 100` ``` by (fact subset_eq) ``` haftmann@31807 ` 101` haftmann@31807 ` 102` ```lemma subset: ``` haftmann@31807 ` 103` ``` "A \ B \ A \ B \ \ B \ A" ``` haftmann@31807 ` 104` ``` by (fact less_le_not_le) ``` haftmann@31807 ` 105` haftmann@31807 ` 106` ```lemma set_eq: ``` haftmann@31807 ` 107` ``` "A = B \ A \ B \ B \ A" ``` haftmann@31807 ` 108` ``` by (fact eq_iff) ``` haftmann@31807 ` 109` haftmann@31807 ` 110` ```lemma inter: ``` haftmann@31807 ` 111` ``` "A \ B = project (\x. x \ A) B" ``` haftmann@31807 ` 112` ``` by (auto simp add: project_def) ``` haftmann@31807 ` 113` haftmann@31807 ` 114` `end`