src/HOL/Library/Crude_Executable_Set.thy

+(*  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 bot_def [symmetric] 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 top_def [symmetric] 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