src/HOL/Library/Dlist.thy

tuned notation, proofs, namespace

(* Author: Florian Haftmann, TU Muenchen *)

section \<open>Lists with elements distinct as canonical example for datatype invariants\<close>

theory Dlist
imports Main
begin

subsection \<open>The type of distinct lists\<close>

typedef '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 \<open>Formal, totalized constructor for @{typ "'a dlist"}:\<close>

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 \<open>Fundamental operations:\<close>

context
begin

qualified definition empty :: "'a dlist" where
  "empty = Dlist []"

qualified definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
  "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"

qualified definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
  "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"

qualified definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where
  "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"

qualified definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
  "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"

end


text \<open>Derived operations:\<close>

context
begin

qualified definition null :: "'a dlist \<Rightarrow> bool" where
  "null dxs = List.null (list_of_dlist dxs)"

qualified definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where
  "member dxs = List.member (list_of_dlist dxs)"

qualified definition length :: "'a dlist \<Rightarrow> nat" where
  "length dxs = List.length (list_of_dlist dxs)"

qualified 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)"

qualified 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)"

end


subsection \<open>Executable version obeying invariant\<close>

lemma list_of_dlist_empty [simp, code abstract]:
  "list_of_dlist Dlist.empty = []"
  by (simp add: Dlist.empty_def)

lemma list_of_dlist_insert [simp, code abstract]:
  "list_of_dlist (Dlist.insert x dxs) = List.insert x (list_of_dlist dxs)"
  by (simp add: Dlist.insert_def)

lemma list_of_dlist_remove [simp, code abstract]:
  "list_of_dlist (Dlist.remove x dxs) = remove1 x (list_of_dlist dxs)"
  by (simp add: Dlist.remove_def)

lemma list_of_dlist_map [simp, code abstract]:
  "list_of_dlist (Dlist.map f dxs) = remdups (List.map f (list_of_dlist dxs))"
  by (simp add: Dlist.map_def)

lemma list_of_dlist_filter [simp, code abstract]:
  "list_of_dlist (Dlist.filter P dxs) = List.filter P (list_of_dlist dxs)"
  by (simp add: Dlist.filter_def)


text \<open>Explicit executable conversion\<close>

definition dlist_of_list [simp]:
  "dlist_of_list = Dlist"

lemma [code abstract]:
  "list_of_dlist (dlist_of_list xs) = remdups xs"
  by simp


text \<open>Equality\<close>

instantiation dlist :: (equal) equal
begin

definition "HOL.equal dxs dys \<longleftrightarrow> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"

instance
  by standard (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 \<open>Induction principle and case distinction\<close>

lemma dlist_induct [case_names empty insert, induct type: dlist]:
  assumes empty: "P Dlist.empty"
  assumes insrt: "\<And>x dxs. \<not> Dlist.member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (Dlist.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 \<open>distinct xs\<close> have "P (Dlist xs)"
  proof (induct xs)
    case Nil from empty show ?case by (simp add: Dlist.empty_def)
  next
    case (Cons x xs)
    then have "\<not> Dlist.member (Dlist xs) x" and "P (Dlist xs)"
      by (simp_all add: Dlist.member_def List.member_def)
    with insrt have "P (Dlist.insert x (Dlist xs))" .
    with Cons show ?case by (simp add: Dlist.insert_def distinct_remdups_id)
  qed
  with dxs show "P dxs" by simp
qed

lemma dlist_case [cases type: dlist]:
  obtains (empty) "dxs = Dlist.empty"
    | (insert) x dys where "\<not> Dlist.member dys x" and "dxs = Dlist.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 = Dlist.empty" by (simp add: Dlist.empty_def) 
    with empty show ?thesis .
  next
    case (Cons x xs)
    with dxs distinct have "\<not> Dlist.member (Dlist xs) x"
      and "dxs = Dlist.insert x (Dlist xs)"
      by (simp_all add: Dlist.member_def List.member_def Dlist.insert_def distinct_remdups_id)
    with insert show ?thesis .
  qed
qed


subsection \<open>Functorial structure\<close>

functor map: map
  by (simp_all add: remdups_map_remdups fun_eq_iff dlist_eq_iff)


subsection \<open>Quickcheck generators\<close>

quickcheck_generator dlist predicate: distinct constructors: Dlist.empty, Dlist.insert

end