--- a/src/HOL/Library/More_Set.thy Tue Dec 27 15:37:33 2011 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,284 +0,0 @@
-
-(* Author: Florian Haftmann, TU Muenchen *)
-
-header {* Relating (finite) sets and lists *}
-
-theory More_Set
-imports Main More_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 [code_unfold_post]:
- "{x \<in> A. P x} = Set.project P A"
- by (simp add: project_def)
-
-
-subsection {* Basic set operations *}
-
-lemma is_empty_set:
- "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
- by (simp add: Set.is_empty_def null_def)
-
-lemma empty_set:
- "{} = 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:
- "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)
-qed
-
-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)
-qed
-
-lemma minus_set:
- "A - set xs = fold Set.remove xs A"
-proof -
- interpret comp_fun_idem Set.remove
- by (fact comp_fun_idem_remove)
- show ?thesis
- by (simp add: minus_fold_remove [of _ A] fold_set)
-qed
-
-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)
-qed
-
-
-subsection {* Derived set operations *}
-
-lemma member:
- "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"
- by simp
-
-lemma subset_eq:
- "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
- by (fact subset_eq)
-
-lemma subset:
- "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
- by (fact less_le_not_le)
-
-lemma set_eq:
- "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
- by (fact eq_iff)
-
-lemma inter:
- "A \<inter> B = Set.project (\<lambda>x. x \<in> A) B"
- by (auto simp add: project_def)
-
-
-subsection {* Theorems on relations *}
-
-lemma product_code:
- "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
- by (auto simp add: Product_Type.product_def)
-
-lemma Id_on_set:
- "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
- by (auto simp add: Id_on_def)
-
-lemma trancl_set_ntrancl: "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
- by (simp add: finite_trancl_ntranl)
-
-lemma set_rel_comp:
- "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)
-
-lemma wf_set:
- "wf (set xs) = acyclic (set xs)"
- by (simp add: wf_iff_acyclic_if_finite)
-
-
-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"
- by (simp_all add: member_def)
-
-lemma [code_unfold]:
- "A = {} \<longleftrightarrow> Set.is_empty A"
- by (simp add: Set.is_empty_def)
-
-declare empty_set [code]
-
-declare is_empty_set [code]
-
-lemma UNIV_coset [code]:
- "UNIV = coset []"
- by simp
-
-lemma insert_code [code]:
- "insert x (set xs) = set (List.insert x xs)"
- "insert x (coset xs) = 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)"
- by (simp_all add: remove_def Compl_insert)
-
-declare image_set [code]
-
-declare project_set [code]
-
-lemma Ball_set [code]:
- "Ball (set xs) P \<longleftrightarrow> list_all P xs"
- by (simp add: list_all_iff)
-
-lemma Bex_set [code]:
- "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
- by (simp add: list_ex_iff)
-
-lemma card_set [code]:
- "card (set xs) = length (remdups xs)"
-proof -
- have "card (set (remdups xs)) = length (remdups xs)"
- by (rule distinct_card) simp
- then show ?thesis by simp
-qed
-
-
-subsection {* Derived operations *}
-
-declare subset_eq [code]
-
-declare subset [code]
-
-
-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)
-
-lemma subtract_code [code]:
- "A - set xs = foldr Set.remove xs A"
- "A - 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)"
- by (auto simp add: union_set_foldr)
-
-definition Inf :: "'a::complete_lattice set \<Rightarrow> 'a" where
- [simp]: "Inf = Complete_Lattices.Inf"
-
-hide_const (open) Inf
-
-lemma [code_unfold_post]:
- "Inf = More_Set.Inf"
- by simp
-
-definition Sup :: "'a::complete_lattice set \<Rightarrow> 'a" where
- [simp]: "Sup = Complete_Lattices.Sup"
-
-hide_const (open) Sup
-
-lemma [code_unfold_post]:
- "Sup = More_Set.Sup"
- by simp
-
-lemma Inf_code [code]:
- "More_Set.Inf (set xs) = foldr inf xs top"
- "More_Set.Inf (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"
- by (simp_all add: Sup_set_foldr)
-
-lemma INF_code [code]:
- "INFI (set xs) f = foldr (inf \<circ> f) xs top"
- by (simp add: INF_def Inf_set_foldr image_set foldr_map del: set_map)
-
-lemma SUP_sup [code]:
- "SUPR (set xs) f = foldr (sup \<circ> f) xs bot"
- by (simp add: SUP_def Sup_set_foldr image_set foldr_map del: set_map)
-
-hide_const (open) coset
-
-
-subsection {* Operations on relations *}
-
-text {* Initially contributed by Tjark Weber. *}
-
-declare Domain_fst [code]
-
-declare Range_snd [code]
-
-declare trans_join [code]
-
-declare irrefl_distinct [code]
-
-declare trancl_set_ntrancl [code]
-
-declare acyclic_irrefl [code]
-
-declare product_code [code]
-
-declare Id_on_set [code]
-
-declare set_rel_comp [code]
-
-declare wf_set [code]
-
-end
-