Theory Dlist

theory Dlist
imports Main
(* Author: Florian Haftmann, TU Muenchen *)

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

theory Dlist
imports Main

subsection {* The type of distinct lists *}

typedef 'a dlist = "{xs::'a list. distinct xs}"
  morphisms list_of_dlist Abs_dlist
  show "[] ∈ {xs. distinct xs}" by simp

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 ( 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 ( 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

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)


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)
    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)
  with dxs show "P dxs" by simp

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 .
    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 .

subsection {* Functorial structure *}

functor map: map
  by (simp_all add: 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