src/HOL/More_Set.thy

 header {* Relating (finite) sets and lists *}
 
 theory More_Set
 imports List
 begin
-
-lemma comp_fun_idem_remove:
-  "comp_fun_idem Set.remove"
-proof -
-  have rem: "Set.remove = (\<lambda>x A. A - {x})" by (simp add: fun_eq_iff remove_def)
-  show ?thesis by (simp only: comp_fun_idem_remove rem)
-qed
-
-lemma minus_fold_remove:
-  assumes "finite A"
-  shows "B - A = Finite_Set.fold Set.remove B A"
-proof -
-  have rem: "Set.remove = (\<lambda>x A. A - {x})" by (simp add: fun_eq_iff remove_def)
-  show ?thesis by (simp only: rem assms minus_fold_remove)
-qed
-
-lemma bounded_Collect_code: (* FIXME delete candidate *)
-  "{x \<in> A. P x} = Set.project P A"
-  by (simp add: project_def)
-
 
 subsection {* Basic set operations *}
 
 lemma is_empty_set [code]:
   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"

 
 lemma empty_set [code]:
   "{} = set []"
   by simp
 
-lemma insert_set_compl:
-  "insert x (- set xs) = - set (removeAll x xs)"
-  by auto
-
-lemma remove_set_compl:
-  "Set.remove x (- set xs) = - set (List.insert x xs)"
-  by (auto simp add: remove_def List.insert_def)
-
-lemma image_set:
-  "image f (set xs) = set (map f xs)"
-  by simp
-
-lemma project_set:
-  "Set.project P (set xs) = set (filter P xs)"
-  by (auto simp add: project_def)
-
 
 subsection {* Functorial set operations *}
 
-lemma union_set:
+lemma union_set_fold:
   "set xs \<union> A = fold Set.insert xs A"
 proof -
   interpret comp_fun_idem Set.insert
     by (fact comp_fun_idem_insert)
   show ?thesis by (simp add: union_fold_insert fold_set_fold)

 lemma union_set_foldr:
   "set xs \<union> A = foldr Set.insert xs A"
 proof -
   have "\<And>x y :: 'a. insert y \<circ> insert x = insert x \<circ> insert y"
     by auto
-  then show ?thesis by (simp add: union_set foldr_fold)
+  then show ?thesis by (simp add: union_set_fold foldr_fold)
 qed
 
-lemma minus_set:
+lemma minus_set_fold:
   "A - set xs = fold Set.remove xs A"
 proof -
   interpret comp_fun_idem Set.remove
     by (fact comp_fun_idem_remove)
   show ?thesis

 lemma minus_set_foldr:
   "A - set xs = foldr Set.remove xs A"
 proof -
   have "\<And>x y :: 'a. Set.remove y \<circ> Set.remove x = Set.remove x \<circ> Set.remove y"
     by (auto simp add: remove_def)
-  then show ?thesis by (simp add: minus_set foldr_fold)
+  then show ?thesis by (simp add: minus_set_fold foldr_fold)
 qed
-
-
-subsection {* Derived set operations *}
-
-lemma member [code]:
-  "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"
-  by simp
-
-lemma subset [code]:
-  "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
-  by (fact less_le_not_le)
-
-lemma set_eq [code]:
-  "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
-  by (fact eq_iff)
-
-lemma inter [code]:
-  "A \<inter> B = Set.project (\<lambda>x. x \<in> A) B"
-  by (auto simp add: project_def)
-
-
-subsection {* Code generator setup *}
-
-definition coset :: "'a list \<Rightarrow> 'a set" where
-  [simp]: "coset xs = - set xs"
-
-code_datatype set coset
 
 
 subsection {* Basic operations *}
 
 lemma [code]:
   "x \<in> set xs \<longleftrightarrow> List.member xs x"
-  "x \<in> coset xs \<longleftrightarrow> \<not> List.member xs x"
+  "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
   by (simp_all add: member_def)
 
 lemma UNIV_coset [code]:
-  "UNIV = coset []"
+  "UNIV = List.coset []"
   by simp
 
 lemma insert_code [code]:
   "insert x (set xs) = set (List.insert x xs)"
-  "insert x (coset xs) = coset (removeAll x xs)"
+  "insert x (List.coset xs) = List.coset (removeAll x xs)"
   by simp_all
 
 lemma remove_code [code]:
   "Set.remove x (set xs) = set (removeAll x xs)"
-  "Set.remove x (coset xs) = coset (List.insert x xs)"
+  "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
   by (simp_all add: remove_def Compl_insert)
 
 lemma Ball_set [code]:
   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
   by (simp add: list_all_iff)

 
 subsection {* Functorial operations *}
 
 lemma inter_code [code]:
   "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
-  "A \<inter> coset xs = foldr Set.remove xs A"
-  by (simp add: inter project_def) (simp add: Diff_eq [symmetric] minus_set_foldr)
+  "A \<inter> List.coset xs = foldr Set.remove xs A"
+  by (simp add: inter_project project_def) (simp add: Diff_eq [symmetric] minus_set_foldr)
 
 lemma subtract_code [code]:
   "A - set xs = foldr Set.remove xs A"
-  "A - coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
+  "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
   by (auto simp add: minus_set_foldr)
 
 lemma union_code [code]:
   "set xs \<union> A = foldr insert xs A"
-  "coset xs \<union> A = coset (List.filter (\<lambda>x. x \<notin> A) xs)"
+  "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
   by (auto simp add: union_set_foldr)
 
 definition Inf :: "'a::complete_lattice set \<Rightarrow> 'a" where
   [simp, code_abbrev]: "Inf = Complete_Lattices.Inf"
 

 
 hide_const (open) Sup
 
 lemma Inf_code [code]:
   "More_Set.Inf (set xs) = foldr inf xs top"
-  "More_Set.Inf (coset []) = bot"
+  "More_Set.Inf (List.coset []) = bot"
   by (simp_all add: Inf_set_foldr)
 
 lemma Sup_sup [code]:
   "More_Set.Sup (set xs) = foldr sup xs bot"
-  "More_Set.Sup (coset []) = top"
+  "More_Set.Sup (List.coset []) = top"
   by (simp_all add: Sup_set_foldr)
 
 (* FIXME: better implement conversion by bisection *)
 
 lemma pred_of_set_fold_sup:

 
 lemma Id_on_set [code]:
   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
   by (auto simp add: Id_on_def)
 
-lemma trancl_set_ntrancl [code]: "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
+lemma trancl_set_ntrancl [code]:
+  "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
   by (simp add: finite_trancl_ntranl)
 
 lemma set_rel_comp [code]:
   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
   by (auto simp add: Bex_def)