(* Author: Tobias Nipkow *)
section \<open>A 2-3 Tree Implementation of Maps\<close>
theory Tree23_Map
imports
Tree23_Set
Map_by_Ordered
begin
fun lookup :: "('a::linorder * 'b) tree23 \<Rightarrow> 'a \<Rightarrow> 'b option" where
"lookup Leaf x = None" |
"lookup (Node2 l (a,b) r) x =
(if x < a then lookup l x else
if a < x then lookup r x else Some b)" |
"lookup (Node3 l (a1,b1) m (a2,b2) r) x =
(if x < a1 then lookup l x else
if x = a1 then Some b1 else
if x < a2 then lookup m x else
if x = a2 then Some b2
else lookup r x)"
fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>i" where
"upd a b Leaf = Up\<^sub>i Leaf (a,b) Leaf" |
"upd a b (Node2 l xy r) =
(if a < fst xy then
(case upd a b l of
T\<^sub>i l' => T\<^sub>i (Node2 l' xy r)
| Up\<^sub>i l1 q l2 => T\<^sub>i (Node3 l1 q l2 xy r))
else if a = fst xy then T\<^sub>i (Node2 l (a,b) r)
else
(case upd a b r of
T\<^sub>i r' => T\<^sub>i (Node2 l xy r')
| Up\<^sub>i r1 q r2 => T\<^sub>i (Node3 l xy r1 q r2)))" |
"upd a b (Node3 l xy1 m xy2 r) =
(if a < fst xy1 then
(case upd a b l of
T\<^sub>i l' => T\<^sub>i (Node3 l' xy1 m xy2 r)
| Up\<^sub>i l1 q l2 => Up\<^sub>i (Node2 l1 q l2) xy1 (Node2 m xy2 r))
else if a = fst xy1 then T\<^sub>i (Node3 l (a,b) m xy2 r)
else if a < fst xy2 then
(case upd a b m of
T\<^sub>i m' => T\<^sub>i (Node3 l xy1 m' xy2 r)
| Up\<^sub>i m1 q m2 => Up\<^sub>i (Node2 l xy1 m1) q (Node2 m2 xy2 r))
else if a = fst xy2 then T\<^sub>i (Node3 l xy1 m (a,b) r)
else
(case upd a b r of
T\<^sub>i r' => T\<^sub>i (Node3 l xy1 m xy2 r')
| Up\<^sub>i r1 q r2 => Up\<^sub>i (Node2 l xy1 m) xy2 (Node2 r1 q r2)))"
definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
"update a b t = tree\<^sub>i(upd a b t)"
fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>d"
where
"del k Leaf = T\<^sub>d Leaf" |
"del k (Node2 Leaf p Leaf) = (if k=fst p then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf p Leaf))" |
"del k (Node3 Leaf p Leaf q Leaf) = T\<^sub>d(if k=fst p then Node2 Leaf q Leaf
else if k=fst q then Node2 Leaf p Leaf else Node3 Leaf p Leaf q Leaf)" |
"del k (Node2 l a r) = (if k<fst a then node21 (del k l) a r else
if k > fst a then node22 l a (del k r) else
let (a',t) = del_min r in node22 l a' t)" |
"del k (Node3 l a m b r) = (if k<fst a then node31 (del k l) a m b r else
if k = fst a then let (a',m') = del_min m in node32 l a' m' b r else
if k < fst b then node32 l a (del k m) b r else
if k = fst b then let (b',r') = del_min r in node33 l a m b' r'
else node33 l a m b (del k r))"
definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
"delete k t = tree\<^sub>d(del k t)"
subsection \<open>Functional Correctness\<close>
lemma lookup: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
by (induction t) (auto simp: map_of_simps split: option.split)
lemma inorder_upd:
"sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)"
by(induction t) (auto simp: upd_list_simps split: up\<^sub>i.splits)
corollary inorder_update:
"sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
by(simp add: update_def inorder_upd)
lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
by(induction t rule: del.induct)
(auto simp: del_list_simps inorder_nodes del_minD split: prod.splits)
corollary inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
inorder(delete x t) = del_list x (inorder t)"
by(simp add: delete_def inorder_del)
subsection \<open>Balancedness\<close>
lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd a b t)) \<and> height(upd a b t) = height t"
by (induct t) (auto split: up\<^sub>i.split)(* 30 secs in 2015 *)
corollary bal_update: "bal t \<Longrightarrow> bal (update a b t)"
by (simp add: update_def bal_upd)
lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
by(induction x t rule: del.induct)
(auto simp add: heights max_def height_del_min split: prod.split)
lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
by(induction x t rule: del.induct)
(auto simp: bals bal_del_min height_del height_del_min split: prod.split)
corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
by(simp add: delete_def bal_tree\<^sub>d_del)
subsection \<open>Overall Correctness\<close>
interpretation T23_Map: Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
and inorder = inorder and wf = bal
proof (standard, goal_cases)
case 2 thus ?case by(simp add: lookup)
next
case 3 thus ?case by(simp add: inorder_update)
next
case 4 thus ?case by(simp add: inorder_delete)
next
case 6 thus ?case by(simp add: bal_update)
next
case 7 thus ?case by(simp add: bal_delete)
qed simp+
end