# Theory DList

theory DList
imports Quotient_List
```(*  Title:      HOL/Quotient_Examples/DList.thy
Author:     Cezary Kaliszyk, University of Tsukuba

Based on typedef version "Library/Dlist" by Florian Haftmann
and theory morphism version by Maksym Bortin
*)

section ‹Lists with distinct elements›

theory DList
imports "HOL-Library.Quotient_List"
begin

lemma remdups_removeAll_commute[simp]:
"remdups (removeAll x l) = removeAll x (remdups l)"
by (induct l) auto

lemma removeAll_distinct[simp]:
assumes "distinct l"
shows "distinct (removeAll x l)"
using assms by (induct l) simp_all

lemma removeAll_commute:
"removeAll x (removeAll y l) = removeAll y (removeAll x l)"
by (induct l) auto

lemma remdups_hd_notin_tl:
"remdups dl = h # t ⟹ h ∉ set t"
proof (induct dl arbitrary: h t)
case Nil
then show ?case by simp
next
case (Cons a dl)
then show ?case by (cases "a ∈ set dl") auto
qed

lemma remdups_repeat:
"remdups dl = h # t ⟹ t = remdups t"
proof (induct dl arbitrary: h t)
case Nil
then show ?case by simp
next
case (Cons a dl)
then show ?case by (cases "a ∈ set dl") (auto simp: remdups_remdups)
qed

lemma remdups_nil_noteq_cons:
"remdups (h # t) ≠ remdups []"
"remdups [] ≠ remdups (h # t)"
by auto

lemma remdups_eq_map_eq:
assumes "remdups xa = remdups ya"
shows "remdups (map f xa) = remdups (map f ya)"
using assms
by (induct xa ya rule: list_induct2')
(metis (full_types) remdups_nil_noteq_cons(2) remdups_map_remdups)+

lemma remdups_eq_member_eq:
assumes "remdups xa = remdups ya"
shows "List.member xa = List.member ya"
using assms
unfolding fun_eq_iff List.member_def
by (induct xa ya rule: list_induct2')
(metis remdups_nil_noteq_cons set_remdups)+

text ‹Setting up the quotient type›

definition
dlist_eq :: "'a list ⇒ 'a list ⇒ bool"
where
[simp]: "dlist_eq xs ys ⟷ remdups xs = remdups ys"

lemma dlist_eq_reflp:
"reflp dlist_eq"
by (auto intro: reflpI)

lemma dlist_eq_symp:
"symp dlist_eq"
by (auto intro: sympI)

lemma dlist_eq_transp:
"transp dlist_eq"
by (auto intro: transpI)

lemma dlist_eq_equivp:
"equivp dlist_eq"
by (auto intro: equivpI dlist_eq_reflp dlist_eq_symp dlist_eq_transp)

quotient_type
'a dlist = "'a list" / "dlist_eq"
by (rule dlist_eq_equivp)

text ‹respectfulness and constant definitions›

definition [simp]: "card_remdups = length ∘ remdups"
definition [simp]: "foldr_remdups f xs e = foldr f (remdups xs) e"

quotient_definition empty where "empty :: 'a dlist"
is "Nil" .

quotient_definition insert where "insert :: 'a ⇒ 'a dlist ⇒ 'a dlist"
is "Cons" by (metis (mono_tags) List.insert_def dlist_eq_def remdups.simps(2) set_remdups)

quotient_definition "member :: 'a dlist ⇒ 'a ⇒ bool"
is "List.member" by (metis dlist_eq_def remdups_eq_member_eq)

quotient_definition foldr where "foldr :: ('a ⇒ 'b ⇒ 'b) ⇒ 'a dlist ⇒ 'b ⇒ 'b"
is "foldr_remdups" by auto

quotient_definition "remove :: 'a ⇒ 'a dlist ⇒ 'a dlist"
is "removeAll" by force

quotient_definition card where "card :: 'a dlist ⇒ nat"
is "card_remdups" by fastforce

quotient_definition map where "map :: ('a ⇒ 'b) ⇒ 'a dlist ⇒ 'b dlist"
is "List.map :: ('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" by (metis dlist_eq_def remdups_eq_map_eq)

quotient_definition filter where "filter :: ('a ⇒ bool) ⇒ 'a dlist ⇒ 'a dlist"
is "List.filter" by (metis dlist_eq_def remdups_filter)

quotient_definition "list_of_dlist :: 'a dlist ⇒ 'a list"
is "remdups" by simp

text ‹lifted theorems›

lemma dlist_member_insert:
"member dl x ⟹ insert x dl = dl"

lemma dlist_member_insert_eq:
"member (insert y dl) x = (x = y ∨ member dl x)"

lemma dlist_insert_idem:
"insert x (insert x dl) = insert x dl"
by descending simp

lemma dlist_insert_not_empty:
"insert x dl ≠ empty"
by descending auto

lemma not_dlist_member_empty:
"¬ member empty x"

lemma not_dlist_member_remove:
"¬ member (remove x dl) x"

lemma dlist_in_remove:
"a ≠ b ⟹ member (remove b dl) a = member dl a"

lemma dlist_not_memb_remove:
"¬ member dl x ⟹ remove x dl = dl"

lemma dlist_no_memb_remove_insert:
"¬ member dl x ⟹ remove x (insert x dl) = dl"

lemma dlist_remove_empty:
"remove x empty = empty"
by descending simp

lemma dlist_remove_insert_commute:
"a ≠ b ⟹ remove a (insert b dl) = insert b (remove a dl)"
by descending simp

lemma dlist_remove_commute:
"remove a (remove b dl) = remove b (remove a dl)"
by (lifting removeAll_commute)

lemma dlist_foldr_empty:
"foldr f empty e = e"
by descending simp

lemma dlist_no_memb_foldr:
assumes "¬ member dl x"
shows "foldr f (insert x dl) e = f x (foldr f dl e)"
using assms by descending (simp add: List.member_def)

lemma dlist_foldr_insert_not_memb:
assumes "¬member t h"
shows "foldr f (insert h t) e = f h (foldr f t e)"
using assms by descending (simp add: List.member_def)

lemma list_of_dlist_empty[simp]:
"list_of_dlist empty = []"
by descending simp

lemma list_of_dlist_insert[simp]:
"list_of_dlist (insert x xs) =
(if member xs x then (remdups (list_of_dlist xs))
else x # (remdups (list_of_dlist xs)))"
by descending (simp add: List.member_def remdups_remdups)

lemma list_of_dlist_remove[simp]:
"list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"

lemma list_of_dlist_map[simp]:
"list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"

lemma list_of_dlist_filter [simp]:
"list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"

lemma dlist_map_empty:
"map f empty = empty"
by descending simp

lemma dlist_map_insert:
"map f (insert x xs) = insert (f x) (map f xs)"
by descending simp

lemma dlist_eq_iff:
"dxs = dys ⟷ list_of_dlist dxs = list_of_dlist dys"
by descending simp

lemma dlist_eqI:
"list_of_dlist dxs = list_of_dlist dys ⟹ dxs = dys"

abbreviation
"dlist xs ≡ abs_dlist xs"

lemma distinct_list_of_dlist [simp, intro]:
"distinct (list_of_dlist dxs)"
by descending simp

lemma list_of_dlist_dlist [simp]:
"list_of_dlist (dlist xs) = remdups xs"
unfolding list_of_dlist_def map_fun_apply id_def
by (metis Quotient3_rep_abs[OF Quotient3_dlist] dlist_eq_def)

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"
(metis Quotient3_def Quotient3_dlist dlist_eqI list_of_dlist_dlist remdups_list_of_dlist)

lemma dlist_filter_simps:
"filter P empty = empty"
"filter P (insert x xs) = (if P x then insert x (filter P xs) else filter P xs)"
by (lifting filter.simps)

lemma dlist_induct:
assumes "P empty"
and "⋀a dl. P dl ⟹ P (insert a dl)"
shows "P dl"
using assms by descending (drule list.induct, simp)

lemma dlist_induct_stronger:
assumes a1: "P empty"
and     a2: "⋀x dl. ⟦¬member dl x; P dl⟧ ⟹ P (insert x dl)"
shows "P dl"
proof(induct dl rule: dlist_induct)
show "P empty" using a1 by simp
next
fix x dl
assume "P dl"
then show "P (insert x dl)" using a2
by (cases "member dl x") (simp_all add: dlist_member_insert)
qed

lemma dlist_card_induct:
assumes "⋀xs. (⋀ys. card ys < card xs ⟹ P ys) ⟹ P xs"
shows "P xs"
using assms
by descending (rule measure_induct [of card_remdups], blast)

lemma dlist_cases:
"dl = empty ∨ (∃h t. dl = insert h t ∧ ¬ member t h)"
by descending
(metis List.member_def dlist_eq_def hd_Cons_tl list_of_dlist_dlist remdups_hd_notin_tl remdups_list_of_dlist)

lemma dlist_exhaust:
obtains "y = empty" | a dl where "y = insert a dl"
by (lifting list.exhaust)

lemma dlist_exhaust_stronger:
obtains "y = empty" | a dl where "y = insert a dl" "¬ member dl a"
by (metis dlist_cases)

end
```