--- a/src/HOL/Library/Executable_Set.thy Mon Dec 07 09:21:14 2009 +0100
+++ b/src/HOL/Library/Executable_Set.thy Mon Dec 07 14:54:01 2009 +0100
@@ -1,103 +1,271 @@
(* Title: HOL/Library/Executable_Set.thy
Author: Stefan Berghofer, TU Muenchen
+ Author: Florian Haftmann, TU Muenchen
*)
-header {* Implementation of finite sets by lists *}
+header {* A crude implementation of finite sets by lists -- avoid using this at any cost! *}
theory Executable_Set
-imports Main Fset SML_Quickcheck
+imports List_Set
begin
-subsection {* Preprocessor setup *}
+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
-declare member [code]
+consts_code
+ Coset ("\<module>Coset")
+ Set ("\<module>Set")
+attach {*
+ datatype 'a set = Set of 'a list | Coset of 'a list;
+*} -- {* This assumes that there won't be a @{text Coset} without a @{text Set} *}
+
+
+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_unfold]:
+ "A = {} \<longleftrightarrow> is_empty A"
+ by simp
definition empty :: "'a set" where
- "empty = {}"
+ [simp]: "empty = {}"
+
+lemma [code_unfold]:
+ "{} = empty"
+ by simp
+
+lemma [code_unfold, code_inline del]:
+ "empty = Set []"
+ by simp -- {* Otherwise @{text \<eta>}-expansion produces funny things. *}
+
+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)
-declare empty_def [symmetric, code_unfold]
+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_unfold]:
+ "op = = set_eq"
+ by simp
+
+definition subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
+ [simp]: "subset_eq = op \<subseteq>"
+
+lemma [code_unfold]:
+ "op \<subseteq> = subset_eq"
+ by simp
+
+definition subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
+ [simp]: "subset = op \<subset>"
+
+lemma [code_unfold]:
+ "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
- "inter = op \<inter>"
+ [simp]: "inter = op \<inter>"
+
+lemma [code_unfold]:
+ "op \<inter> = inter"
+ by simp
-declare inter_def [symmetric, code_unfold]
+definition subtract :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
+ [simp]: "subtract A B = B - A"
+
+lemma [code_unfold]:
+ "B - A = subtract A B"
+ by simp
definition union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
- "union = op \<union>"
-
-declare union_def [symmetric, code_unfold]
+ [simp]: "union = op \<union>"
-definition subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
- "subset = op \<le>"
+lemma [code_unfold]:
+ "op \<union> = union"
+ by simp
-declare subset_def [symmetric, code_unfold]
-
-lemma [code]:
- "subset A B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
- by (simp add: subset_def subset_eq)
+definition Inf :: "'a::complete_lattice set \<Rightarrow> 'a" where
+ [simp]: "Inf = Complete_Lattice.Inf"
-definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
- [code del]: "eq_set = op ="
-
-(*declare eq_set_def [symmetric, code_unfold]*)
+lemma [code_unfold]:
+ "Complete_Lattice.Inf = Inf"
+ by simp
-lemma [code]:
- "eq_set A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
- by (simp add: eq_set_def set_eq)
+definition Sup :: "'a::complete_lattice set \<Rightarrow> 'a" where
+ [simp]: "Sup = Complete_Lattice.Sup"
-declare inter [code]
-
-declare List_Set.project_def [symmetric, code_unfold]
+lemma [code_unfold]:
+ "Complete_Lattice.Sup = Sup"
+ by simp
definition Inter :: "'a set set \<Rightarrow> 'a set" where
- "Inter = Complete_Lattice.Inter"
+ [simp]: "Inter = Inf"
-declare Inter_def [symmetric, code_unfold]
+lemma [code_unfold]:
+ "Inf = Inter"
+ by simp
definition Union :: "'a set set \<Rightarrow> 'a set" where
- "Union = Complete_Lattice.Union"
+ [simp]: "Union = Sup"
-declare Union_def [symmetric, code_unfold]
-
+lemma [code_unfold]:
+ "Sup = Union"
+ by simp
-subsection {* Code generator setup *}
-
-ML {*
-nonfix inter;
-nonfix union;
-nonfix subset;
+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")
*}
-definition flip :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
- "flip f a b = f b a"
+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)
-types_code
- fset ("(_/ \<module>fset)")
-attach {*
-datatype 'a fset = Set of 'a list | Coset of 'a list;
-*}
+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)
-consts_code
- Set ("\<module>Set")
- Coset ("\<module>Coset")
+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)
-consts_code
- "empty" ("{*Fset.empty*}")
- "List_Set.is_empty" ("{*Fset.is_empty*}")
- "Set.insert" ("{*Fset.insert*}")
- "List_Set.remove" ("{*Fset.remove*}")
- "Set.image" ("{*Fset.map*}")
- "List_Set.project" ("{*Fset.filter*}")
- "Ball" ("{*flip Fset.forall*}")
- "Bex" ("{*flip Fset.exists*}")
- "union" ("{*Fset.union*}")
- "inter" ("{*Fset.inter*}")
- "op - \<Colon> 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" ("{*flip Fset.subtract*}")
- "Union" ("{*Fset.Union*}")
- "Inter" ("{*Fset.Inter*}")
- card ("{*Fset.card*}")
- fold ("{*foldl o flip*}")
+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
-hide (open) const empty inter union subset eq_set Inter Union flip
+lemma Union_union [code]:
+ "Union (Set xs) = foldl union empty xs"
+ "Union (Coset []) = Coset []"
+ unfolding Union_def Sup_sup by simp_all
-end
\ No newline at end of file
+hide (open) const is_empty empty remove
+ set_eq subset_eq subset inter union subtract Inf Sup Inter Union
+
+end