(* Author: Florian Haftmann, TU Muenchen *)

 header {* Lists with elements distinct as canonical example for datatype invariants *}

 theory Dlist

 imports Main

 begin

 subsection {* The type of distinct lists *}

 typedef (open) 'a dlist = "{xs::'a list. distinct xs}"

   morphisms list_of_dlist Abs_dlist

 proof

   show "[] \<in> {xs. distinct xs}" 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 = 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)"

 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 \<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

 declare equal_dlist_def [code]

 lemma [code nbe]:

   "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> 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: "\<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

 subsection {* Functorial structure *}

 enriched_type 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