--- a/src/HOL/Library/Crude_Executable_Set.thy Tue Dec 08 18:38:08 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,259 +0,0 @@
-(* Title: HOL/Library/Crude_Executable_Set.thy
- Author: Florian Haftmann, TU Muenchen
-*)
-
-header {* A crude implementation of finite sets by lists -- avoid using this at any cost! *}
-
-theory Crude_Executable_Set
-imports List_Set
-begin
-
-declare mem_def [code del]
-declare Collect_def [code del]
-declare insert_code [code del]
-declare vimage_code [code del]
-
-subsection {* Set representation *}
-
-setup {*
- Code.add_type_cmd "set"
-*}
-
-definition Set :: "'a list \<Rightarrow> 'a set" where
- [simp]: "Set = set"
-
-definition Coset :: "'a list \<Rightarrow> 'a set" where
- [simp]: "Coset xs = - set xs"
-
-setup {*
- Code.add_signature_cmd ("Set", "'a list \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("Coset", "'a list \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("set", "'a list \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("op \<in>", "'a \<Rightarrow> 'a set \<Rightarrow> bool")
-*}
-
-code_datatype Set Coset
-
-
-subsection {* Basic operations *}
-
-lemma [code]:
- "set xs = Set (remdups xs)"
- by simp
-
-lemma [code]:
- "x \<in> Set xs \<longleftrightarrow> member x xs"
- "x \<in> Coset xs \<longleftrightarrow> \<not> member x xs"
- by (simp_all add: mem_iff)
-
-definition is_empty :: "'a set \<Rightarrow> bool" where
- [simp]: "is_empty A \<longleftrightarrow> A = {}"
-
-lemma [code_inline]:
- "A = {} \<longleftrightarrow> is_empty A"
- by simp
-
-definition empty :: "'a set" where
- [simp]: "empty = {}"
-
-lemma [code_inline]:
- "{} = empty"
- by simp
-
-setup {*
- Code.add_signature_cmd ("is_empty", "'a set \<Rightarrow> bool")
- #> Code.add_signature_cmd ("empty", "'a set")
- #> Code.add_signature_cmd ("insert", "'a \<Rightarrow> 'a set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("List_Set.remove", "'a \<Rightarrow> 'a set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("image", "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set")
- #> Code.add_signature_cmd ("List_Set.project", "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("Ball", "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool")
- #> Code.add_signature_cmd ("Bex", "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool")
- #> Code.add_signature_cmd ("card", "'a set \<Rightarrow> nat")
-*}
-
-lemma is_empty_Set [code]:
- "is_empty (Set xs) \<longleftrightarrow> null xs"
- by (simp add: empty_null)
-
-lemma empty_Set [code]:
- "empty = Set []"
- by simp
-
-lemma insert_Set [code]:
- "insert x (Set xs) = Set (List_Set.insert x xs)"
- "insert x (Coset xs) = Coset (remove_all x xs)"
- by (simp_all add: insert_set insert_set_compl)
-
-lemma remove_Set [code]:
- "remove x (Set xs) = Set (remove_all x xs)"
- "remove x (Coset xs) = Coset (List_Set.insert x xs)"
- by (simp_all add:remove_set remove_set_compl)
-
-lemma image_Set [code]:
- "image f (Set xs) = Set (remdups (map f xs))"
- by simp
-
-lemma project_Set [code]:
- "project P (Set xs) = Set (filter P xs)"
- by (simp add: project_set)
-
-lemma Ball_Set [code]:
- "Ball (Set xs) P \<longleftrightarrow> list_all P xs"
- by (simp add: ball_set)
-
-lemma Bex_Set [code]:
- "Bex (Set xs) P \<longleftrightarrow> list_ex P xs"
- by (simp add: bex_set)
-
-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 *}
-
-definition set_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
- [simp]: "set_eq = op ="
-
-lemma [code_inline]:
- "op = = set_eq"
- by simp
-
-definition subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
- [simp]: "subset_eq = op \<subseteq>"
-
-lemma [code_inline]:
- "op \<subseteq> = subset_eq"
- by simp
-
-definition subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
- [simp]: "subset = op \<subset>"
-
-lemma [code_inline]:
- "op \<subset> = subset"
- by simp
-
-setup {*
- Code.add_signature_cmd ("set_eq", "'a set \<Rightarrow> 'a set \<Rightarrow> bool")
- #> Code.add_signature_cmd ("subset_eq", "'a set \<Rightarrow> 'a set \<Rightarrow> bool")
- #> Code.add_signature_cmd ("subset", "'a set \<Rightarrow> 'a set \<Rightarrow> bool")
-*}
-
-lemma set_eq_subset_eq [code]:
- "set_eq A B \<longleftrightarrow> subset_eq A B \<and> subset_eq B A"
- by auto
-
-lemma subset_eq_forall [code]:
- "subset_eq A B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
- by (simp add: subset_eq)
-
-lemma subset_subset_eq [code]:
- "subset A B \<longleftrightarrow> subset_eq A B \<and> \<not> subset_eq B A"
- by (simp add: subset)
-
-
-subsection {* Functorial operations *}
-
-definition inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
- [simp]: "inter = op \<inter>"
-
-lemma [code_inline]:
- "op \<inter> = inter"
- by simp
-
-definition subtract :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
- [simp]: "subtract A B = B - A"
-
-lemma [code_inline]:
- "B - A = subtract A B"
- by simp
-
-definition union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
- [simp]: "union = op \<union>"
-
-lemma [code_inline]:
- "op \<union> = union"
- by simp
-
-definition Inf :: "'a::complete_lattice set \<Rightarrow> 'a" where
- [simp]: "Inf = Complete_Lattice.Inf"
-
-lemma [code_inline]:
- "Complete_Lattice.Inf = Inf"
- by simp
-
-definition Sup :: "'a::complete_lattice set \<Rightarrow> 'a" where
- [simp]: "Sup = Complete_Lattice.Sup"
-
-lemma [code_inline]:
- "Complete_Lattice.Sup = Sup"
- by simp
-
-definition Inter :: "'a set set \<Rightarrow> 'a set" where
- [simp]: "Inter = Inf"
-
-lemma [code_inline]:
- "Inf = Inter"
- by simp
-
-definition Union :: "'a set set \<Rightarrow> 'a set" where
- [simp]: "Union = Sup"
-
-lemma [code_inline]:
- "Sup = Union"
- by simp
-
-setup {*
- Code.add_signature_cmd ("inter", "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("subtract", "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("union", "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("Inf", "'a set \<Rightarrow> 'a")
- #> Code.add_signature_cmd ("Sup", "'a set \<Rightarrow> 'a")
- #> Code.add_signature_cmd ("Inter", "'a set set \<Rightarrow> 'a set")
- #> Code.add_signature_cmd ("Union", "'a set set \<Rightarrow> 'a set")
-*}
-
-lemma inter_project [code]:
- "inter A (Set xs) = Set (List.filter (\<lambda>x. x \<in> A) xs)"
- "inter A (Coset xs) = foldl (\<lambda>A x. remove x A) A xs"
- by (simp add: inter project_def, simp add: Diff_eq [symmetric] minus_set)
-
-lemma subtract_remove [code]:
- "subtract (Set xs) A = foldl (\<lambda>A x. remove x A) A xs"
- "subtract (Coset xs) A = Set (List.filter (\<lambda>x. x \<in> A) xs)"
- by (auto simp add: minus_set)
-
-lemma union_insert [code]:
- "union (Set xs) A = foldl (\<lambda>A x. insert x A) A xs"
- "union (Coset xs) A = Coset (List.filter (\<lambda>x. x \<notin> A) xs)"
- by (auto simp add: union_set)
-
-lemma Inf_inf [code]:
- "Inf (Set xs) = foldl inf (top :: 'a::complete_lattice) xs"
- "Inf (Coset []) = (bot :: 'a::complete_lattice)"
- by (simp_all add: Inf_UNIV Inf_set_fold)
-
-lemma Sup_sup [code]:
- "Sup (Set xs) = foldl sup (bot :: 'a::complete_lattice) xs"
- "Sup (Coset []) = (top :: 'a::complete_lattice)"
- by (simp_all add: Sup_UNIV Sup_set_fold)
-
-lemma Inter_inter [code]:
- "Inter (Set xs) = foldl inter (Coset []) xs"
- "Inter (Coset []) = empty"
- unfolding Inter_def Inf_inf by simp_all
-
-lemma Union_union [code]:
- "Union (Set xs) = foldl union empty xs"
- "Union (Coset []) = Coset []"
- unfolding Union_def Sup_sup by simp_all
-
-hide (open) const is_empty empty remove
- set_eq subset_eq subset inter union subtract Inf Sup Inter Union
-
-end