(* Author: Florian Haftmann, TU Muenchen *)

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

theory Dlist

imports Main More_List Fset

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_ext:

  assumes "list_of_dlist xs = list_of_dlist ys"

  shows "xs = ys"

  using assms by (simp add: list_of_dlist_inject)

text {* Formal, totalized constructor for @{typ "'a dlist"}: *}

definition Dlist :: "'a list \<Rightarrow> 'a dlist" where

  [code del]: "Dlist xs = Abs_dlist (remdups xs)"

lemma distinct_list_of_dlist [simp]:

  "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 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

section {* Implementation of sets by distinct lists -- canonical! *}

definition Set :: "'a dlist \<Rightarrow> 'a fset" where

  "Set dxs = Fset.Set (list_of_dlist dxs)"

definition Coset :: "'a dlist \<Rightarrow> 'a fset" where

  "Coset dxs = Fset.Coset (list_of_dlist dxs)"

code_datatype Set Coset

declare member_code [code del]

declare is_empty_Set [code del]

declare empty_Set [code del]

declare UNIV_Set [code del]

declare insert_Set [code del]

declare remove_Set [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) = Fset (set xs)"

  by (simp add: Set_def Fset.Set_def)

lemma Coset_Dlist [simp]:

  "Coset (Dlist xs) = Fset (- set xs)"

  by (simp add: Coset_def Fset.Coset_def)

lemma member_Set [simp]:

  "Fset.member (Set dxs) = List.member (list_of_dlist dxs)"

  by (simp add: Set_def member_set)

lemma member_Coset [simp]:

  "Fset.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]:

  "Fset.Set xs = Set (dlist_of_list xs)"

  by simp

lemma Coset_dlist_of_list [code]:

  "Fset.Coset xs = Coset (dlist_of_list xs)"

  by simp

lemma is_empty_Set [code]:

  "Fset.is_empty (Set dxs) \<longleftrightarrow> null dxs"

  by (simp add: null_def null_empty 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]:

  "Fset.insert x (Set dxs) = Set (insert x dxs)"

  "Fset.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]:

  "Fset.remove x (Set dxs) = Set (remove x dxs)"

  "Fset.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]:

  "Fset.member (Set dxs) = member dxs"

  "Fset.member (Coset dxs) = Not \<circ> member dxs"

  by (simp_all add: member_def)

lemma map_code [code]:

  "Fset.map f (Set dxs) = Set (map f dxs)"

  by (simp add: member_set)

lemma filter_code [code]:

  "Fset.filter f (Set dxs) = Set (filter f dxs)"

  by (simp add: member_set)

lemma forall_Set [code]:

  "Fset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)"

  by (simp add: member_set list_all_iff)

lemma exists_Set [code]:

  "Fset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)"

  by (simp add: member_set list_ex_iff)

lemma card_code [code]:

  "Fset.card (Set dxs) = length dxs"

  by (simp add: length_def member_set distinct_card)

lemma inter_code [code]:

  "inf A (Set xs) = Set (filter (Fset.member A) xs)"

  "inf A (Coset xs) = foldr Fset.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 Fset.remove xs A"

  "A - Coset xs = Set (filter (Fset.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 Fset.insert xs A"

  "sup (Coset xs) A = Coset (filter (Not \<circ> Fset.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