(* Author: Florian Haftmann, TU Muenchen *)
header {* Lists with elements distinct as canonical example for datatype invariants *}
theory Dlist
imports Main 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_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 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
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
lemma [code nbe]:
"HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True"
by (fact equal_refl)
section {* 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 rule: distinct_induct)
case Nil from empty show ?case by (simp add: empty_def)
next
case (insert 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 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 \<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
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 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 (rule fset_eqI) (simp add: Set_def)
lemma Coset_Dlist [simp]:
"Coset (Dlist xs) = Fset (- set xs)"
by (rule fset_eqI) (simp add: 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 (rule fset_eqI) simp
lemma Coset_dlist_of_list [code]:
"Fset.Coset xs = Coset (dlist_of_list xs)"
by (rule fset_eqI) simp
lemma is_empty_Set [code]:
"Fset.is_empty (Set dxs) \<longleftrightarrow> null dxs"
by (simp add: null_def List.null_def 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 compl_code [code]:
"- Set dxs = Coset dxs"
"- Coset dxs = Set dxs"
by (rule fset_eqI, simp add: member_set not_set_compl)+
lemma map_code [code]:
"Fset.map f (Set dxs) = Set (map f dxs)"
by (rule fset_eqI) (simp add: member_set)
lemma filter_code [code]:
"Fset.filter f (Set dxs) = Set (filter f dxs)"
by (rule fset_eqI) (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