(* Author: Florian Haftmann, TU Muenchen *)
header {* Canonical implementation of sets by distinct lists *}
theory Dlist_Cset
imports Dlist List_Cset
begin
definition Set :: "'a dlist \<Rightarrow> 'a Cset.set" where
"Set dxs = List_Cset.set (list_of_dlist dxs)"
definition Coset :: "'a dlist \<Rightarrow> 'a Cset.set" where
"Coset dxs = List_Cset.coset (list_of_dlist dxs)"
code_datatype Set Coset
declare member_code [code del]
declare List_Cset.is_empty_set [code del]
declare List_Cset.empty_set [code del]
declare List_Cset.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) = Cset.Set (set xs)"
by (rule Cset.set_eqI) (simp add: Set_def)
lemma Coset_Dlist [simp]:
"Coset (Dlist xs) = Cset.Set (- set xs)"
by (rule Cset.set_eqI) (simp add: Coset_def)
lemma member_Set [simp]:
"Cset.member (Set dxs) = List.member (list_of_dlist dxs)"
by (simp add: Set_def member_set)
lemma member_Coset [simp]:
"Cset.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]:
"List_Cset.set xs = Set (dlist_of_list xs)"
by (rule Cset.set_eqI) simp
lemma Coset_dlist_of_list [code]:
"List_Cset.coset xs = Coset (dlist_of_list xs)"
by (rule Cset.set_eqI) simp
lemma is_empty_Set [code]:
"Cset.is_empty (Set dxs) \<longleftrightarrow> Dlist.null dxs"
by (simp add: Dlist.null_def List.null_def member_set)
lemma bot_code [code]:
"bot = Set Dlist.empty"
by (simp add: empty_def)
lemma top_code [code]:
"top = Coset Dlist.empty"
by (simp add: empty_def)
lemma insert_code [code]:
"Cset.insert x (Set dxs) = Set (Dlist.insert x dxs)"
"Cset.insert x (Coset dxs) = Coset (Dlist.remove x dxs)"
by (simp_all add: Dlist.insert_def Dlist.remove_def member_set not_set_compl)
lemma remove_code [code]:
"Cset.remove x (Set dxs) = Set (Dlist.remove x dxs)"
"Cset.remove x (Coset dxs) = Coset (Dlist.insert x dxs)"
by (auto simp add: Dlist.insert_def Dlist.remove_def member_set not_set_compl)
lemma member_code [code]:
"Cset.member (Set dxs) = Dlist.member dxs"
"Cset.member (Coset dxs) = Not \<circ> Dlist.member dxs"
by (simp_all add: member_def)
lemma compl_code [code]:
"- Set dxs = Coset dxs"
"- Coset dxs = Set dxs"
by (rule Cset.set_eqI, simp add: member_set not_set_compl)+
lemma map_code [code]:
"Cset.map f (Set dxs) = Set (Dlist.map f dxs)"
by (rule Cset.set_eqI) (simp add: member_set)
lemma filter_code [code]:
"Cset.filter f (Set dxs) = Set (Dlist.filter f dxs)"
by (rule Cset.set_eqI) (simp add: member_set)
lemma forall_Set [code]:
"Cset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)"
by (simp add: member_set list_all_iff)
lemma exists_Set [code]:
"Cset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)"
by (simp add: member_set list_ex_iff)
lemma card_code [code]:
"Cset.card (Set dxs) = Dlist.length dxs"
by (simp add: length_def member_set distinct_card)
lemma inter_code [code]:
"inf A (Set xs) = Set (Dlist.filter (Cset.member A) xs)"
"inf A (Coset xs) = Dlist.foldr Cset.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 = Dlist.foldr Cset.remove xs A"
"A - Coset xs = Set (Dlist.filter (Cset.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 = Dlist.foldr Cset.insert xs A"
"sup (Coset xs) A = Coset (Dlist.filter (Not \<circ> Cset.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) = Dlist.foldr inf As top"
by (simp only: Set_def Infimum_inf foldr_def inf.commute)
lemma Supremum_code [code]:
"Supremum (Set As) = Dlist.foldr sup As bot"
by (simp only: Set_def Supremum_sup foldr_def sup.commute)
end
end