(* Author: Florian Haftmann, TU Muenchen *)
section \<open>Lists with elements distinct as canonical example for datatype invariants\<close>
theory Dlist
imports Main
begin
subsection \<open>The type of distinct lists\<close>
typedef 'a dlist = "{xs::'a list. distinct xs}"
morphisms list_of_dlist Abs_dlist
proof
show "[] \<in> {xs. distinct xs}" by simp
qed
lemma dlist_eq_iff:
"dxs = dys \<longleftrightarrow> 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 \<Longrightarrow> dxs = dys"
by (simp add: dlist_eq_iff)
text \<open>Formal, totalized constructor for @{typ "'a dlist"}:\<close>
definition Dlist :: "'a list \<Rightarrow> '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 \<open>Fundamental operations:\<close>
definition empty :: "'a dlist" where
"empty = Dlist []"
definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
"insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"
definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
"remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where
"map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"
definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
"filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"
text \<open>Derived operations:\<close>
definition null :: "'a dlist \<Rightarrow> bool" where
"null dxs = List.null (list_of_dlist dxs)"
definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where
"member dxs = List.member (list_of_dlist dxs)"
definition length :: "'a dlist \<Rightarrow> nat" where
"length dxs = List.length (list_of_dlist dxs)"
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
"fold f dxs = List.fold f (list_of_dlist dxs)"
definition foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
"foldr f dxs = List.foldr f (list_of_dlist dxs)"
subsection \<open>Executable version obeying invariant\<close>
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 \<open>Explicit executable conversion\<close>
definition dlist_of_list [simp]:
"dlist_of_list = Dlist"
lemma [code abstract]:
"list_of_dlist (dlist_of_list xs) = remdups xs"
by simp
text \<open>Equality\<close>
instantiation dlist :: (equal) equal
begin
definition "HOL.equal dxs dys \<longleftrightarrow> 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 \<longleftrightarrow> True"
by (fact equal_refl)
subsection \<open>Induction principle and case distinction\<close>
lemma dlist_induct [case_names empty insert, induct type: dlist]:
assumes empty: "P empty"
assumes insrt: "\<And>x dxs. \<not> member dxs x \<Longrightarrow> P dxs \<Longrightarrow> 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 \<open>distinct xs\<close> 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 "\<not> 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 [cases type: dlist]:
obtains (empty) "dxs = empty"
| (insert) x dys where "\<not> member dys x" and "dxs = 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 = empty" by (simp add: empty_def)
with empty show ?thesis .
next
case (Cons x xs)
with dxs distinct have "\<not> 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 ?thesis .
qed
qed
subsection \<open>Functorial structure\<close>
functor map: map
by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff)
subsection \<open>Quickcheck generators\<close>
quickcheck_generator dlist predicate: distinct constructors: empty, insert
hide_const (open) member fold foldr empty insert remove map filter null member length fold
end