(* Author: Florian Haftmann, TU Muenchen *)

header {* 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

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: *}

definition empty :: "'a dlist" where

"empty = Dlist []"

definition insert :: "'a => 'a dlist => 'a dlist" where

"insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"

definition remove :: "'a => 'a dlist => 'a dlist" where

"remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"

definition map :: "('a => 'b) => 'a dlist => 'b dlist" where

"map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"

definition filter :: "('a => bool) => 'a dlist => 'a dlist" where

"filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"

text {* Derived operations: *}

definition null :: "'a dlist => bool" where

"null dxs = List.null (list_of_dlist dxs)"

definition member :: "'a dlist => 'a => bool" where

"member dxs = List.member (list_of_dlist dxs)"

definition length :: "'a dlist => nat" where

"length dxs = List.length (list_of_dlist dxs)"

definition fold :: "('a => 'b => 'b) => 'a dlist => 'b => 'b" where

"fold f dxs = List.fold f (list_of_dlist dxs)"

definition foldr :: "('a => 'b => 'b) => 'a dlist => 'b => 'b" where

"foldr f dxs = List.foldr f (list_of_dlist dxs)"

subsection {* 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 <-> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"

instance proof

qed (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 empty"

assumes insrt: "!!x dxs. ¬ member dxs x ==> P dxs ==> 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)

case Nil from empty show ?case by (simp add: empty_def)

next

case (Cons x xs)

then have "¬ 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 Cons 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 ==> P"

assumes insert: "!!x dys. ¬ member dys x ==> dxs = insert x dys ==> 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 "¬ 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

subsection {* Functorial structure *}

enriched_type map: map

by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff)

subsection {* Quickcheck generators *}

quickcheck_generator dlist predicate: distinct constructors: empty, insert

hide_const (open) member fold foldr empty insert remove map filter null member length fold

end