# Theory Dlist

theory Dlist
imports Main
```(* Author: Florian Haftmann, TU Muenchen
Author: Andreas Lochbihler, ETH Zurich *)

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

theory Dlist
imports Main
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

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 fun wpull :: "('a × 'b) list ⇒ ('b × 'c) list ⇒ ('a × 'c) list"
where
"wpull [] ys = []"
| "wpull xs [] = []"
| "wpull ((a, b) # xs) ((b', c) # ys) =
(if b ∈ snd ` set xs then
(a, the (map_of (rev ((b', c) # ys)) b)) # wpull xs ((b', c) # ys)
else if b' ∈ fst ` set ys then
(the (map_of (map prod.swap (rev ((a, b) # xs))) b'), c) # wpull ((a, b) # xs) ys
else (a, c) # wpull xs ys)"

qualified lemma wpull_eq_Nil_iff [simp]: "wpull xs ys = [] ⟷ xs = [] ∨ ys = []"
by(cases "(xs, ys)" rule: wpull.cases) simp_all

qualified lemma wpull_induct
[consumes 1,
case_names Nil left[xs eq in_set IH] right[xs ys eq in_set IH] step[xs ys eq IH] ]:
assumes eq: "remdups (map snd xs) = remdups (map fst ys)"
and Nil: "P [] []"
and left: "⋀a b xs b' c ys.
⟦ b ∈ snd ` set xs; remdups (map snd xs) = remdups (map fst ((b', c) # ys));
(b, the (map_of (rev ((b', c) # ys)) b)) ∈ set ((b', c) # ys); P xs ((b', c) # ys) ⟧
⟹ P ((a, b) # xs) ((b', c) # ys)"
and right: "⋀a b xs b' c ys.
⟦ b ∉ snd ` set xs; b' ∈ fst ` set ys;
remdups (map snd ((a, b) # xs)) = remdups (map fst ys);
(the (map_of (map prod.swap (rev ((a, b) #xs))) b'), b') ∈ set ((a, b) # xs);
P ((a, b) # xs) ys ⟧
⟹ P ((a, b) # xs) ((b', c) # ys)"
and step: "⋀a b xs c ys.
⟦ b ∉ snd ` set xs; b ∉ fst ` set ys; remdups (map snd xs) = remdups (map fst ys);
P xs ys ⟧
⟹ P ((a, b) # xs) ((b, c) # ys)"
shows "P xs ys"
using eq
proof(induction xs ys rule: wpull.induct)
case 1 thus ?case by(simp add: Nil)
next
case 2 thus ?case by(simp split: if_split_asm)
next
case Cons: (3 a b xs b' c ys)
let ?xs = "(a, b) # xs" and ?ys = "(b', c) # ys"
consider (xs) "b ∈ snd ` set xs" | (ys) "b ∉ snd ` set xs" "b' ∈ fst ` set ys"
| (step) "b ∉ snd ` set xs" "b' ∉ fst ` set ys" by auto
thus ?case
proof cases
case xs
with Cons.prems have eq: "remdups (map snd xs) = remdups (map fst ?ys)" by auto
from xs eq have "b ∈ fst ` set ?ys" by (metis list.set_map set_remdups)
hence "map_of (rev ?ys) b ≠ None" unfolding map_of_eq_None_iff by auto
then obtain c' where "map_of (rev ?ys) b = Some c'" by blast
then have "(b, the (map_of (rev ?ys) b)) ∈ set ?ys" by(auto dest: map_of_SomeD split: if_split_asm)
from xs eq this Cons.IH(1)[OF xs eq] show ?thesis by(rule left)
next
case ys
from ys Cons.prems have eq: "remdups (map snd ?xs) = remdups (map fst ys)" by auto
from ys eq have "b' ∈ snd ` set ?xs" by (metis list.set_map set_remdups)
hence "map_of (map prod.swap (rev ?xs)) b' ≠ None"
unfolding map_of_eq_None_iff by(auto simp add: image_image)
then obtain a' where "map_of (map prod.swap (rev ?xs)) b' = Some a'" by blast
then have "(the (map_of (map prod.swap (rev ?xs)) b'), b') ∈ set ?xs"
by(auto dest: map_of_SomeD split: if_split_asm)
from ys eq this Cons.IH(2)[OF ys eq] show ?thesis by(rule right)
next
case *: step
hence "remdups (map snd xs) = remdups (map fst ys)" "b = b'" using Cons.prems by auto
from * this(1) Cons.IH(3)[OF * this(1)] show ?thesis unfolding ‹b = b'› by(rule step)
qed
qed

qualified lemma set_wpull_subset:
assumes "remdups (map snd xs) = remdups (map fst ys)"
shows "set (wpull xs ys) ⊆ set xs O set ys"
using assms by(induction xs ys rule: wpull_induct) auto

qualified lemma set_fst_wpull:
assumes "remdups (map snd xs) = remdups (map fst ys)"
shows "fst ` set (wpull xs ys) = fst ` set xs"
using assms by(induction xs ys rule: wpull_induct)(auto intro: rev_image_eqI)

qualified lemma set_snd_wpull:
assumes "remdups (map snd xs) = remdups (map fst ys)"
shows "snd ` set (wpull xs ys) = snd ` set ys"
using assms by(induction xs ys rule: wpull_induct)(auto intro: rev_image_eqI)

qualified lemma wpull:
assumes "distinct xs"
and "distinct ys"
and "set xs ⊆ {(x, y). R x y}"
and "set ys ⊆ {(x, y). S x y}"
and eq: "remdups (map snd xs) = remdups (map fst ys)"
shows "∃zs. distinct zs ∧ set zs ⊆ {(x, y). (R OO S) x y} ∧
remdups (map fst zs) = remdups (map fst xs) ∧ remdups (map snd zs) = remdups (map snd ys)"
proof(intro exI conjI)
let ?zs = "remdups (wpull xs ys)"
show "distinct ?zs" by simp
show "set ?zs ⊆ {(x, y). (R OO S) x y}" using assms(3-4) set_wpull_subset[OF eq] by fastforce
show "remdups (map fst ?zs) = remdups (map fst xs)" unfolding remdups_map_remdups using eq
by(induction xs ys rule: wpull_induct)(auto simp add: set_fst_wpull intro: rev_image_eqI)
show "remdups (map snd ?zs) = remdups (map snd ys)" unfolding remdups_map_remdups using eq
by(induction xs ys rule: wpull_induct)(auto simp add: set_snd_wpull intro: rev_image_eqI)
qed

qualified lift_definition set :: "'a dlist ⇒ 'a set" is List.set .

qualified lemma map_transfer [transfer_rule]:
"(rel_fun op = (rel_fun (pcr_dlist op =) (pcr_dlist op =))) (λf x. remdups (List.map f x)) Dlist.map"
by(simp add: rel_fun_def dlist.pcr_cr_eq cr_dlist_def Dlist.map_def remdups_remdups)

bnf "'a dlist"
map: Dlist.map
sets: set
bd: natLeq
wits: Dlist.empty
unfolding OO_Grp_alt mem_Collect_eq
subgoal by(rule ext)(simp add: dlist_eq_iff)
subgoal by(rule ext)(simp add: dlist_eq_iff remdups_map_remdups)
subgoal by(simp add: dlist_eq_iff set_def cong: list.map_cong)
subgoal by(simp add: set_def fun_eq_iff)
subgoal by(simp add: natLeq_card_order)
subgoal by(simp add: natLeq_cinfinite)
subgoal by(rule ordLess_imp_ordLeq)(simp add: finite_iff_ordLess_natLeq[symmetric] set_def)
subgoal by(rule predicate2I)(transfer; auto simp add: wpull)
subgoal by(simp add: set_def)
done

lifting_update dlist.lifting
lifting_forget dlist.lifting

end

end
```