# Theory Dlist

```(* Author: Florian Haftmann, TU Muenchen
Author: Andreas Lochbihler, ETH Zurich *)

section ‹Lists with elements distinct as canonical example for datatype invariants›

theory Dlist
imports Confluent_Quotient
begin

subsection ‹The type of distinct lists›

typedef 'a dlist = "{xs::'a list. distinct xs}"
morphisms list_of_dlist Abs_dlist
proof
show "[] ∈ {xs. distinct xs}" by simp
qed

context begin

qualified definition dlist_eq where "dlist_eq = BNF_Def.vimage2p remdups remdups (=)"

qualified lemma equivp_dlist_eq: "equivp dlist_eq"
unfolding dlist_eq_def by(rule equivp_vimage2p)(rule identity_equivp)

qualified definition abs_dlist :: "'a list ⇒ 'a dlist" where "abs_dlist = Abs_dlist o remdups"

definition qcr_dlist :: "'a list ⇒ 'a dlist ⇒ bool" where "qcr_dlist x y ⟷ y = abs_dlist x"

qualified lemma Quotient_dlist_remdups: "Quotient dlist_eq abs_dlist list_of_dlist qcr_dlist"
unfolding Quotient_def dlist_eq_def qcr_dlist_def vimage2p_def abs_dlist_def
by (auto simp add: fun_eq_iff Abs_dlist_inject
list_of_dlist[simplified] list_of_dlist_inverse distinct_remdups_id)

end

locale Quotient_dlist begin
setup_lifting Dlist.Quotient_dlist_remdups Dlist.equivp_dlist_eq[THEN equivp_reflp2]
end

setup_lifting type_definition_dlist

lemma dlist_eq_iff:
"dxs = dys ⟷ 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 ⟹ dxs = dys"
by (simp add: dlist_eq_iff)

text ‹Formal, totalized constructor for \<^typ>‹'a dlist›:›

definition Dlist :: "'a list ⇒ '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 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"
by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id)

text ‹Fundamental operations:›

context
begin

qualified definition empty :: "'a dlist" where
"empty = Dlist []"

qualified definition insert :: "'a ⇒ 'a dlist ⇒ 'a dlist" where
"insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"

qualified definition remove :: "'a ⇒ 'a dlist ⇒ 'a dlist" where
"remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"

qualified definition map :: "('a ⇒ 'b) ⇒ 'a dlist ⇒ 'b dlist" where
"map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"

qualified definition filter :: "('a ⇒ bool) ⇒ 'a dlist ⇒ 'a dlist" where
"filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"

qualified definition rotate :: "nat ⇒ 'a dlist ⇒ 'a dlist" where
"rotate n dxs = Dlist (List.rotate n (list_of_dlist dxs))"

end

text ‹Derived operations:›

context
begin

qualified definition null :: "'a dlist ⇒ bool" where
"null dxs = List.null (list_of_dlist dxs)"

qualified definition member :: "'a dlist ⇒ 'a ⇒ bool" where
"member dxs = List.member (list_of_dlist dxs)"

qualified definition length :: "'a dlist ⇒ nat" where
"length dxs = List.length (list_of_dlist dxs)"

qualified definition fold :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a dlist ⇒ 'b ⇒ 'b" where
"fold f dxs = List.fold f (list_of_dlist dxs)"

qualified definition foldr :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a dlist ⇒ 'b ⇒ 'b" where
"foldr f dxs = List.foldr f (list_of_dlist dxs)"

end

subsection ‹Executable version obeying invariant›

lemma list_of_dlist_empty [simp, code abstract]:
"list_of_dlist Dlist.empty = []"
by (simp add: Dlist.empty_def)

lemma list_of_dlist_insert [simp, code abstract]:
"list_of_dlist (Dlist.insert x dxs) = List.insert x (list_of_dlist dxs)"
by (simp add: Dlist.insert_def)

lemma list_of_dlist_remove [simp, code abstract]:
"list_of_dlist (Dlist.remove x dxs) = remove1 x (list_of_dlist dxs)"
by (simp add: Dlist.remove_def)

lemma list_of_dlist_map [simp, code abstract]:
"list_of_dlist (Dlist.map f dxs) = remdups (List.map f (list_of_dlist dxs))"
by (simp add: Dlist.map_def)

lemma list_of_dlist_filter [simp, code abstract]:
"list_of_dlist (Dlist.filter P dxs) = List.filter P (list_of_dlist dxs)"
by (simp add: Dlist.filter_def)

lemma list_of_dlist_rotate [simp, code abstract]:
"list_of_dlist (Dlist.rotate n dxs) = List.rotate n (list_of_dlist dxs)"
by (simp add: Dlist.rotate_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 ⟷ HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"

instance
by standard (simp add: equal_dlist_def equal list_of_dlist_inject)

end

declare equal_dlist_def [code]

lemma [code nbe]: "HOL.equal (dxs :: 'a::equal dlist) dxs ⟷ True"
by (fact equal_refl)

subsection ‹Induction principle and case distinction›

lemma dlist_induct [case_names empty insert, induct type: dlist]:
assumes empty: "P Dlist.empty"
assumes insrt: "⋀x dxs. ¬ Dlist.member dxs x ⟹ P dxs ⟹ P (Dlist.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)
case Nil from empty show ?case by (simp add: Dlist.empty_def)
next
case (Cons x xs)
then have "¬ Dlist.member (Dlist xs) x" and "P (Dlist xs)"
by (simp_all add: Dlist.member_def List.member_def)
with insrt have "P (Dlist.insert x (Dlist xs))" .
with Cons show ?case by (simp add: Dlist.insert_def distinct_remdups_id)
qed
with dxs show "P dxs" by simp
qed

lemma dlist_case [cases type: dlist]:
obtains (empty) "dxs = Dlist.empty"
| (insert) x dys where "¬ Dlist.member dys x" and "dxs = Dlist.insert x dys"
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 thesis
proof (cases xs)
case Nil with dxs
have "dxs = Dlist.empty" by (simp add: Dlist.empty_def)
with empty show ?thesis .
next
case (Cons x xs)
with dxs distinct have "¬ Dlist.member (Dlist xs) x"
and "dxs = Dlist.insert x (Dlist xs)"
by (simp_all add: Dlist.member_def List.member_def Dlist.insert_def distinct_remdups_id)
with insert show ?thesis .
qed
qed

subsection ‹Functorial structure›

functor map: map
by (simp_all add: remdups_map_remdups fun_eq_iff dlist_eq_iff)

subsection ‹Quickcheck generators›

quickcheck_generator dlist predicate: distinct constructors: Dlist.empty, Dlist.insert

subsection ‹BNF instance›

context begin

qualified inductive double :: "'a list ⇒ 'a list ⇒ bool" where
"double (xs @ ys) (xs @ x # ys)" if "x ∈ set ys"

qualified lemma strong_confluentp_double: "strong_confluentp double"
proof
fix xs ys zs :: "'a list"
assume ys: "double xs ys" and zs: "double xs zs"
consider (left) as y bs z cs where "xs = as @ bs @ cs" "ys = as @ y # bs @ cs" "zs = as @ bs @ z # cs" "y ∈ set (bs @ cs)" "z ∈ set cs"
| (right) as y bs z cs where "xs = as @ bs @ cs" "ys = as @ bs @ y # cs" "zs = as @ z # bs @ cs" "y ∈ set cs" "z ∈ set (bs @ cs)"
proof -
show thesis using ys zs
by(clarsimp simp add: double.simps append_eq_append_conv2)(auto intro: that)
qed
then show "∃us. double⇧*⇧* ys us ∧ double⇧=⇧= zs us"
proof cases
case left
let ?us = "as @ y # bs @ z # cs"
have "double ys ?us" "double zs ?us" using left
by(auto 4 4 simp add: double.simps)(metis append_Cons append_assoc)+
then show ?thesis by blast
next
case right
let ?us = "as @ z # bs @ y # cs"
have "double ys ?us" "double zs ?us" using right
by(auto 4 4 simp add: double.simps)(metis append_Cons append_assoc)+
then show ?thesis by blast
qed
qed

qualified lemma double_Cons1 [simp]: "double xs (x # xs)" if "x ∈ set xs"
using double.intros[of x xs "[]"] that by simp

qualified lemma double_Cons_same [simp]: "double xs ys ⟹ double (x # xs) (x # ys)"
by(auto simp add: double.simps Cons_eq_append_conv)

qualified lemma doubles_Cons_same: "double⇧*⇧* xs ys ⟹ double⇧*⇧* (x # xs) (x # ys)"
by(induction rule: rtranclp_induct)(auto intro: rtranclp.rtrancl_into_rtrancl)

qualified lemma remdups_into_doubles: "double⇧*⇧* (remdups xs) xs"
by(induction xs)(auto intro: doubles_Cons_same rtranclp.rtrancl_into_rtrancl)

qualified lemma dlist_eq_into_doubles: "Dlist.dlist_eq ≤ equivclp double"
by(auto 4 4 simp add: Dlist.dlist_eq_def vimage2p_def
intro: equivclp_trans converse_rtranclp_into_equivclp rtranclp_into_equivclp remdups_into_doubles)

qualified lemma factor_double_map: "double (map f xs) ys ⟹ ∃zs. Dlist.dlist_eq xs zs ∧ ys = map f zs ∧ set zs ⊆ set xs"
by(auto simp add: double.simps Dlist.dlist_eq_def vimage2p_def map_eq_append_conv)
(metis (no_types, opaque_lifting) list.simps(9) map_append remdups.simps(2) remdups_append2 set_append set_eq_subset set_remdups)

qualified lemma dlist_eq_set_eq: "Dlist.dlist_eq xs ys ⟹ set xs = set ys"
by(simp add: Dlist.dlist_eq_def vimage2p_def)(metis set_remdups)

qualified lemma dlist_eq_map_respect: "Dlist.dlist_eq xs ys ⟹ Dlist.dlist_eq (map f xs) (map f ys)"
by(clarsimp simp add: Dlist.dlist_eq_def vimage2p_def)(metis remdups_map_remdups)

qualified lemma confluent_quotient_dlist:
"confluent_quotient double Dlist.dlist_eq Dlist.dlist_eq Dlist.dlist_eq Dlist.dlist_eq Dlist.dlist_eq
(map fst) (map snd) (map fst) (map snd) list_all2 list_all2 list_all2 set set"
by(unfold_locales)(auto intro: strong_confluentp_imp_confluentp strong_confluentp_double
dest: factor_double_map dlist_eq_into_doubles[THEN predicate2D] dlist_eq_set_eq
simp add: list.in_rel list.rel_compp dlist_eq_map_respect Dlist.equivp_dlist_eq equivp_imp_transp)

lifting_update dlist.lifting
lifting_forget dlist.lifting

end

context begin
interpretation Quotient_dlist: Quotient_dlist .

lift_bnf (plugins del: code) 'a dlist
subgoal for A B by(rule confluent_quotient.subdistributivity[OF Dlist.confluent_quotient_dlist])
subgoal by(force dest: Dlist.dlist_eq_set_eq intro: equivp_reflp[OF Dlist.equivp_dlist_eq])
done

qualified lemma list_of_dlist_transfer[transfer_rule]:
"bi_unique R ⟹ (rel_fun (Quotient_dlist.pcr_dlist R) (list_all2 R)) remdups list_of_dlist"
unfolding rel_fun_def Quotient_dlist.pcr_dlist_def qcr_dlist_def Dlist.abs_dlist_def
by (auto simp: Abs_dlist_inverse intro!: remdups_transfer[THEN rel_funD])

lemma list_of_dlist_map_dlist[simp]:
"list_of_dlist (map_dlist f xs) = remdups (map f (list_of_dlist xs))"
by transfer (auto simp: remdups_map_remdups)

end

end
```