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src/HOL/Library/Dlist.thy

author | wenzelm |

Sun, 30 Jan 2011 13:02:18 +0100 | |

changeset 41648 | 6d736d983d5c |

parent 41505 | 6d19301074cf |

child 43146 | 09f74fda1b1d |

permissions | -rw-r--r-- |

clarified example settings for Proof General;

(* Author: Florian Haftmann, TU Muenchen *) header {* Lists with elements distinct as canonical example for datatype invariants *} theory Dlist imports Main Cset begin section {* The type of distinct lists *} typedef (open) 'a dlist = "{xs::'a list. distinct xs}" morphisms list_of_dlist Abs_dlist proof show "[] \<in> ?dlist" by simp qed lemma dlist_eq_iff: "dxs = dys \<longleftrightarrow> list_of_dlist dxs = list_of_dlist dys" by (simp add: list_of_dlist_inject) lemma dlist_eqI: "list_of_dlist dxs = list_of_dlist dys \<Longrightarrow> dxs = dys" by (simp add: dlist_eq_iff) text {* Formal, totalized constructor for @{typ "'a dlist"}: *} definition Dlist :: "'a list \<Rightarrow> 'a dlist" where "Dlist xs = Abs_dlist (remdups xs)" lemma distinct_list_of_dlist [simp, intro]: "distinct (list_of_dlist dxs)" using list_of_dlist [of dxs] by simp lemma list_of_dlist_Dlist [simp]: "list_of_dlist (Dlist xs) = remdups xs" by (simp add: Dlist_def Abs_dlist_inverse) lemma remdups_list_of_dlist [simp]: "remdups (list_of_dlist dxs) = list_of_dlist dxs" by simp lemma Dlist_list_of_dlist [simp, code abstype]: "Dlist (list_of_dlist dxs) = dxs" by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id) text {* Fundamental operations: *} definition empty :: "'a dlist" where "empty = Dlist []" definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))" definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))" definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))" definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))" text {* Derived operations: *} definition null :: "'a dlist \<Rightarrow> bool" where "null dxs = List.null (list_of_dlist dxs)" definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where "member dxs = List.member (list_of_dlist dxs)" definition length :: "'a dlist \<Rightarrow> nat" where "length dxs = List.length (list_of_dlist dxs)" definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where "fold f dxs = More_List.fold f (list_of_dlist dxs)" definition foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where "foldr f dxs = List.foldr f (list_of_dlist dxs)" section {* Executable version obeying invariant *} lemma list_of_dlist_empty [simp, code abstract]: "list_of_dlist empty = []" by (simp add: empty_def) lemma list_of_dlist_insert [simp, code abstract]: "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)" by (simp add: insert_def) lemma list_of_dlist_remove [simp, code abstract]: "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)" by (simp add: remove_def) lemma list_of_dlist_map [simp, code abstract]: "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))" by (simp add: map_def) lemma list_of_dlist_filter [simp, code abstract]: "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)" by (simp add: filter_def) text {* Explicit executable conversion *} definition dlist_of_list [simp]: "dlist_of_list = Dlist" lemma [code abstract]: "list_of_dlist (dlist_of_list xs) = remdups xs" by simp text {* Equality *} instantiation dlist :: (equal) equal begin definition "HOL.equal dxs dys \<longleftrightarrow> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)" instance proof qed (simp add: equal_dlist_def equal list_of_dlist_inject) end lemma [code nbe]: "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True" by (fact equal_refl) section {* Induction principle and case distinction *} lemma dlist_induct [case_names empty insert, induct type: dlist]: assumes empty: "P empty" assumes insrt: "\<And>x dxs. \<not> member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (insert x dxs)" shows "P dxs" proof (cases dxs) case (Abs_dlist xs) then have "distinct xs" and dxs: "dxs = Dlist xs" by (simp_all add: Dlist_def distinct_remdups_id) from `distinct xs` have "P (Dlist xs)" proof (induct xs) case Nil from empty show ?case by (simp add: empty_def) next case (Cons x xs) then have "\<not> member (Dlist xs) x" and "P (Dlist xs)" by (simp_all add: member_def List.member_def) with insrt have "P (insert x (Dlist xs))" . with Cons show ?case by (simp add: insert_def distinct_remdups_id) qed with dxs show "P dxs" by simp qed lemma dlist_case [case_names empty insert, cases type: dlist]: assumes empty: "dxs = empty \<Longrightarrow> P" assumes insert: "\<And>x dys. \<not> member dys x \<Longrightarrow> dxs = insert x dys \<Longrightarrow> P" shows P proof (cases dxs) case (Abs_dlist xs) then have dxs: "dxs = Dlist xs" and distinct: "distinct xs" by (simp_all add: Dlist_def distinct_remdups_id) show P proof (cases xs) case Nil with dxs have "dxs = empty" by (simp add: empty_def) with empty show P . next case (Cons x xs) with dxs distinct have "\<not> member (Dlist xs) x" and "dxs = insert x (Dlist xs)" by (simp_all add: member_def List.member_def insert_def distinct_remdups_id) with insert show P . qed qed section {* Functorial structure *} enriched_type map: map by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff) section {* Implementation of sets by distinct lists -- canonical! *} definition Set :: "'a dlist \<Rightarrow> 'a Cset.set" where "Set dxs = Cset.set (list_of_dlist dxs)" definition Coset :: "'a dlist \<Rightarrow> 'a Cset.set" where "Coset dxs = Cset.coset (list_of_dlist dxs)" code_datatype Set Coset declare member_code [code del] declare Cset.is_empty_set [code del] declare Cset.empty_set [code del] declare Cset.UNIV_set [code del] declare insert_set [code del] declare remove_set [code del] declare compl_set [code del] declare compl_coset [code del] declare map_set [code del] declare filter_set [code del] declare forall_set [code del] declare exists_set [code del] declare card_set [code del] declare inter_project [code del] declare subtract_remove [code del] declare union_insert [code del] declare Infimum_inf [code del] declare Supremum_sup [code del] lemma Set_Dlist [simp]: "Set (Dlist xs) = Cset.Set (set xs)" by (rule Cset.set_eqI) (simp add: Set_def) lemma Coset_Dlist [simp]: "Coset (Dlist xs) = Cset.Set (- set xs)" by (rule Cset.set_eqI) (simp add: Coset_def) lemma member_Set [simp]: "Cset.member (Set dxs) = List.member (list_of_dlist dxs)" by (simp add: Set_def member_set) lemma member_Coset [simp]: "Cset.member (Coset dxs) = Not \<circ> List.member (list_of_dlist dxs)" by (simp add: Coset_def member_set not_set_compl) lemma Set_dlist_of_list [code]: "Cset.set xs = Set (dlist_of_list xs)" by (rule Cset.set_eqI) simp lemma Coset_dlist_of_list [code]: "Cset.coset xs = Coset (dlist_of_list xs)" by (rule Cset.set_eqI) simp lemma is_empty_Set [code]: "Cset.is_empty (Set dxs) \<longleftrightarrow> null dxs" by (simp add: null_def List.null_def member_set) lemma bot_code [code]: "bot = Set empty" by (simp add: empty_def) lemma top_code [code]: "top = Coset empty" by (simp add: empty_def) lemma insert_code [code]: "Cset.insert x (Set dxs) = Set (insert x dxs)" "Cset.insert x (Coset dxs) = Coset (remove x dxs)" by (simp_all add: insert_def remove_def member_set not_set_compl) lemma remove_code [code]: "Cset.remove x (Set dxs) = Set (remove x dxs)" "Cset.remove x (Coset dxs) = Coset (insert x dxs)" by (auto simp add: insert_def remove_def member_set not_set_compl) lemma member_code [code]: "Cset.member (Set dxs) = member dxs" "Cset.member (Coset dxs) = Not \<circ> member dxs" by (simp_all add: member_def) lemma compl_code [code]: "- Set dxs = Coset dxs" "- Coset dxs = Set dxs" by (rule Cset.set_eqI, simp add: member_set not_set_compl)+ lemma map_code [code]: "Cset.map f (Set dxs) = Set (map f dxs)" by (rule Cset.set_eqI) (simp add: member_set) lemma filter_code [code]: "Cset.filter f (Set dxs) = Set (filter f dxs)" by (rule Cset.set_eqI) (simp add: member_set) lemma forall_Set [code]: "Cset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)" by (simp add: member_set list_all_iff) lemma exists_Set [code]: "Cset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)" by (simp add: member_set list_ex_iff) lemma card_code [code]: "Cset.card (Set dxs) = length dxs" by (simp add: length_def member_set distinct_card) lemma inter_code [code]: "inf A (Set xs) = Set (filter (Cset.member A) xs)" "inf A (Coset xs) = foldr Cset.remove xs A" by (simp_all only: Set_def Coset_def foldr_def inter_project list_of_dlist_filter) lemma subtract_code [code]: "A - Set xs = foldr Cset.remove xs A" "A - Coset xs = Set (filter (Cset.member A) xs)" by (simp_all only: Set_def Coset_def foldr_def subtract_remove list_of_dlist_filter) lemma union_code [code]: "sup (Set xs) A = foldr Cset.insert xs A" "sup (Coset xs) A = Coset (filter (Not \<circ> Cset.member A) xs)" by (simp_all only: Set_def Coset_def foldr_def union_insert list_of_dlist_filter) context complete_lattice begin lemma Infimum_code [code]: "Infimum (Set As) = foldr inf As top" by (simp only: Set_def Infimum_inf foldr_def inf.commute) lemma Supremum_code [code]: "Supremum (Set As) = foldr sup As bot" by (simp only: Set_def Supremum_sup foldr_def sup.commute) end hide_const (open) member fold foldr empty insert remove map filter null member length fold end