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
 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@37024 ` 6` ```theory More_Set ``` haftmann@37023 ` 7` ```imports Main More_List ``` 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@42871 ` 18` ```lemma comp_fun_idem_remove: ``` haftmann@42871 ` 19` ``` "comp_fun_idem remove" ``` haftmann@31807 ` 20` ```proof - ``` nipkow@39302 ` 21` ``` have rem: "remove = (\x A. A - {x})" by (simp add: fun_eq_iff remove_def) ``` haftmann@42871 ` 22` ``` show ?thesis by (simp only: comp_fun_idem_remove rem) ``` haftmann@31807 ` 23` ```qed ``` haftmann@31807 ` 24` haftmann@31807 ` 25` ```lemma minus_fold_remove: ``` haftmann@31807 ` 26` ``` assumes "finite A" ``` haftmann@37023 ` 27` ``` shows "B - A = Finite_Set.fold remove B A" ``` haftmann@31807 ` 28` ```proof - ``` nipkow@39302 ` 29` ``` have rem: "remove = (\x A. A - {x})" by (simp add: fun_eq_iff 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@45012 ` 34` ``` "project P A = {a \ A. P a}" ``` haftmann@45012 ` 35` haftmann@45012 ` 36` ```lemma bounded_Collect_code [code_unfold_post]: ``` haftmann@45012 ` 37` ``` "{x \ A. P x} = project P A" ``` haftmann@45012 ` 38` ``` by (simp add: project_def) ``` haftmann@45012 ` 39` haftmann@45012 ` 40` ```definition product :: "'a set \ 'b set \ ('a \ 'b) set" where ``` haftmann@45012 ` 41` ``` "product A B = Sigma A (\_. B)" ``` haftmann@45012 ` 42` haftmann@45012 ` 43` ```hide_const (open) product ``` haftmann@45012 ` 44` haftmann@45012 ` 45` ```lemma [code_unfold_post]: ``` haftmann@45012 ` 46` ``` "Sigma A (\_. B) = More_Set.product A B" ``` haftmann@45012 ` 47` ``` by (simp add: product_def) ``` haftmann@31807 ` 48` haftmann@31807 ` 49` haftmann@31807 ` 50` ```subsection {* Basic set operations *} ``` haftmann@31807 ` 51` haftmann@31807 ` 52` ```lemma is_empty_set: ``` haftmann@37595 ` 53` ``` "is_empty (set xs) \ List.null xs" ``` haftmann@37595 ` 54` ``` by (simp add: is_empty_def null_def) ``` haftmann@31807 ` 55` haftmann@31807 ` 56` ```lemma empty_set: ``` haftmann@31807 ` 57` ``` "{} = set []" ``` haftmann@31807 ` 58` ``` by simp ``` haftmann@31807 ` 59` haftmann@32880 ` 60` ```lemma insert_set_compl: ``` haftmann@34977 ` 61` ``` "insert x (- set xs) = - set (removeAll x xs)" ``` haftmann@34977 ` 62` ``` by auto ``` haftmann@31807 ` 63` haftmann@32880 ` 64` ```lemma remove_set_compl: ``` haftmann@34977 ` 65` ``` "remove x (- set xs) = - set (List.insert x xs)" ``` haftmann@44326 ` 66` ``` by (auto simp add: remove_def List.insert_def) ``` haftmann@32880 ` 67` haftmann@31807 ` 68` ```lemma image_set: ``` haftmann@31846 ` 69` ``` "image f (set xs) = set (map f xs)" ``` haftmann@31807 ` 70` ``` by simp ``` haftmann@31807 ` 71` haftmann@31807 ` 72` ```lemma project_set: ``` haftmann@31807 ` 73` ``` "project P (set xs) = set (filter P xs)" ``` haftmann@31807 ` 74` ``` by (auto simp add: project_def) ``` haftmann@31807 ` 75` haftmann@31807 ` 76` haftmann@31807 ` 77` ```subsection {* Functorial set operations *} ``` haftmann@31807 ` 78` haftmann@31807 ` 79` ```lemma union_set: ``` haftmann@37023 ` 80` ``` "set xs \ A = fold Set.insert xs A" ``` haftmann@31807 ` 81` ```proof - ``` haftmann@42871 ` 82` ``` interpret comp_fun_idem Set.insert ``` haftmann@42871 ` 83` ``` by (fact comp_fun_idem_insert) ``` haftmann@31807 ` 84` ``` show ?thesis by (simp add: union_fold_insert fold_set) ``` haftmann@31807 ` 85` ```qed ``` haftmann@31807 ` 86` haftmann@37023 ` 87` ```lemma union_set_foldr: ``` haftmann@37023 ` 88` ``` "set xs \ A = foldr Set.insert xs A" ``` haftmann@37023 ` 89` ```proof - ``` haftmann@37023 ` 90` ``` have "\x y :: 'a. insert y \ insert x = insert x \ insert y" ``` haftmann@45012 ` 91` ``` by auto ``` haftmann@37023 ` 92` ``` then show ?thesis by (simp add: union_set foldr_fold) ``` haftmann@37023 ` 93` ```qed ``` haftmann@37023 ` 94` haftmann@31807 ` 95` ```lemma minus_set: ``` haftmann@37023 ` 96` ``` "A - set xs = fold remove xs A" ``` haftmann@31807 ` 97` ```proof - ``` haftmann@42871 ` 98` ``` interpret comp_fun_idem remove ``` haftmann@42871 ` 99` ``` by (fact comp_fun_idem_remove) ``` haftmann@31807 ` 100` ``` show ?thesis ``` haftmann@31807 ` 101` ``` by (simp add: minus_fold_remove [of _ A] fold_set) ``` haftmann@31807 ` 102` ```qed ``` haftmann@31807 ` 103` haftmann@37023 ` 104` ```lemma minus_set_foldr: ``` haftmann@37023 ` 105` ``` "A - set xs = foldr remove xs A" ``` haftmann@37023 ` 106` ```proof - ``` haftmann@37023 ` 107` ``` have "\x y :: 'a. remove y \ remove x = remove x \ remove y" ``` haftmann@45012 ` 108` ``` by (auto simp add: remove_def) ``` haftmann@37023 ` 109` ``` then show ?thesis by (simp add: minus_set foldr_fold) ``` haftmann@37023 ` 110` ```qed ``` haftmann@37023 ` 111` haftmann@31807 ` 112` haftmann@31807 ` 113` ```subsection {* Derived set operations *} ``` haftmann@31807 ` 114` haftmann@31807 ` 115` ```lemma member: ``` haftmann@31807 ` 116` ``` "a \ A \ (\x\A. a = x)" ``` haftmann@31807 ` 117` ``` by simp ``` haftmann@31807 ` 118` haftmann@31807 ` 119` ```lemma subset_eq: ``` haftmann@31807 ` 120` ``` "A \ B \ (\x\A. x \ B)" ``` haftmann@31807 ` 121` ``` by (fact subset_eq) ``` haftmann@31807 ` 122` haftmann@31807 ` 123` ```lemma subset: ``` haftmann@31807 ` 124` ``` "A \ B \ A \ B \ \ B \ A" ``` haftmann@31807 ` 125` ``` by (fact less_le_not_le) ``` haftmann@31807 ` 126` haftmann@31807 ` 127` ```lemma set_eq: ``` haftmann@31807 ` 128` ``` "A = B \ A \ B \ B \ A" ``` haftmann@31807 ` 129` ``` by (fact eq_iff) ``` haftmann@31807 ` 130` haftmann@31807 ` 131` ```lemma inter: ``` haftmann@31807 ` 132` ``` "A \ B = project (\x. x \ A) B" ``` haftmann@31807 ` 133` ``` by (auto simp add: project_def) ``` haftmann@31807 ` 134` haftmann@37023 ` 135` haftmann@45012 ` 136` ```subsection {* Theorems on relations *} ``` haftmann@45012 ` 137` haftmann@45012 ` 138` ```lemma product_code: ``` haftmann@45012 ` 139` ``` "More_Set.product (set xs) (set ys) = set [(x, y). x \ xs, y \ ys]" ``` haftmann@45012 ` 140` ``` by (auto simp add: product_def) ``` haftmann@37023 ` 141` haftmann@45012 ` 142` ```lemma Id_on_set: ``` haftmann@45012 ` 143` ``` "Id_on (set xs) = set [(x, x). x \ xs]" ``` haftmann@45012 ` 144` ``` by (auto simp add: Id_on_def) ``` haftmann@45012 ` 145` haftmann@45012 ` 146` ```lemma set_rel_comp: ``` haftmann@45012 ` 147` ``` "set xys O set yzs = set ([(fst xy, snd yz). xy \ xys, yz \ yzs, snd xy = fst yz])" ``` haftmann@45012 ` 148` ``` by (auto simp add: Bex_def) ``` haftmann@45012 ` 149` haftmann@45012 ` 150` ```lemma wf_set: ``` haftmann@45012 ` 151` ``` "wf (set xs) = acyclic (set xs)" ``` haftmann@45012 ` 152` ``` by (simp add: wf_iff_acyclic_if_finite) ``` haftmann@37023 ` 153` haftmann@37024 ` 154` ```end ```