(* Author: Florian Haftmann, TU Muenchen *) section {* Lists with elements distinct as canonical example for datatype invariants *} theory Dlist imports Main begin subsection {* The type of distinct lists *} typedef 'a dlist = "{xs::'a list. distinct xs}" morphisms list_of_dlist Abs_dlist proof show "[] ∈ {xs. distinct xs}" by simp qed lemma dlist_eq_iff: "dxs = dys <-> 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 ==> dxs = dys" by (simp add: dlist_eq_iff) text {* Formal, totalized constructor for @{typ "'a dlist"}: *} definition Dlist :: "'a list => '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 => 'a dlist => 'a dlist" where "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))" definition remove :: "'a => 'a dlist => 'a dlist" where "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))" definition map :: "('a => 'b) => 'a dlist => 'b dlist" where "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))" definition filter :: "('a => bool) => 'a dlist => 'a dlist" where "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))" text {* Derived operations: *} definition null :: "'a dlist => bool" where "null dxs = List.null (list_of_dlist dxs)" definition member :: "'a dlist => 'a => bool" where "member dxs = List.member (list_of_dlist dxs)" definition length :: "'a dlist => nat" where "length dxs = List.length (list_of_dlist dxs)" definition fold :: "('a => 'b => 'b) => 'a dlist => 'b => 'b" where "fold f dxs = List.fold f (list_of_dlist dxs)" definition foldr :: "('a => 'b => 'b) => 'a dlist => 'b => 'b" where "foldr f dxs = List.foldr f (list_of_dlist dxs)" subsection {* 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 <-> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)" instance proof qed (simp add: equal_dlist_def equal list_of_dlist_inject) end declare equal_dlist_def [code] lemma [code nbe]: "HOL.equal (dxs :: 'a::equal dlist) dxs <-> True" by (fact equal_refl) subsection {* Induction principle and case distinction *} lemma dlist_induct [case_names empty insert, induct type: dlist]: assumes empty: "P empty" assumes insrt: "!!x dxs. ¬ member dxs x ==> P dxs ==> 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 "¬ 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 [cases type: dlist]: obtains (empty) "dxs = empty" | (insert) x dys where "¬ member dys x" and "dxs = insert x dys" 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 thesis proof (cases xs) case Nil with dxs have "dxs = empty" by (simp add: empty_def) with empty show ?thesis . next case (Cons x xs) with dxs distinct have "¬ 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 ?thesis . qed qed subsection {* Functorial structure *} functor map: map by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff) subsection {* Quickcheck generators *} quickcheck_generator dlist predicate: distinct constructors: empty, insert hide_const (open) member fold foldr empty insert remove map filter null member length fold end