Theory Dlist

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theory Dlist
imports Fset

(* 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 "[] ∈ ?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 => '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 => '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 = More_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)"



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 {* 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 rule: distinct_induct)
case Nil from empty show ?case by (simp add: empty_def)
next
case (insert x xs)
then have "¬ member (Dlist xs) x" and "P (Dlist xs)"
by (simp_all add: member_def mem_iff)
with insrt have "P (insert x (Dlist xs))" .
with insert 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 ==> P"
assumes insert: "!!x dys. ¬ member dys x ==> dxs = insert x dys ==> 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 "¬ member (Dlist xs) x"
and "dxs = insert x (Dlist xs)"

by (simp_all add: member_def mem_iff insert_def distinct_remdups_id)
with insert show P .
qed
qed


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

definition Set :: "'a dlist => 'a fset" where
"Set dxs = Fset.Set (list_of_dlist dxs)"


definition Coset :: "'a dlist => '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 compl_Set [code del]
declare compl_Coset [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 o 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) <-> 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 o member dxs"

by (simp_all add: member_def)

lemma compl_code [code]:
"- Set dxs = Coset dxs"
"- Coset dxs = Set dxs"

by (simp_all add: not_set_compl member_set)

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) <-> list_all P (list_of_dlist xs)"

by (simp add: member_set list_all_iff)

lemma exists_Set [code]:
"Fset.exists P (Set xs) <-> 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 o 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