# Theory Tree23_Map

theory Tree23_Map
imports Tree23_Set Map_Specs
```(* Author: Tobias Nipkow *)

section ‹2-3 Tree Implementation of Maps›

theory Tree23_Map
imports
Tree23_Set
Map_Specs
begin

fun lookup :: "('a::linorder * 'b) tree23 ⇒ 'a ⇒ 'b option" where
"lookup Leaf x = None" |
"lookup (Node2 l (a,b) r) x = (case cmp x a of
LT ⇒ lookup l x |
GT ⇒ lookup r x |
EQ ⇒ Some b)" |
"lookup (Node3 l (a1,b1) m (a2,b2) r) x = (case cmp x a1 of
LT ⇒ lookup l x |
EQ ⇒ Some b1 |
GT ⇒ (case cmp x a2 of
LT ⇒ lookup m x |
EQ ⇒ Some b2 |
GT ⇒ lookup r x))"

fun upd :: "'a::linorder ⇒ 'b ⇒ ('a*'b) tree23 ⇒ ('a*'b) up⇩i" where
"upd x y Leaf = Up⇩i Leaf (x,y) Leaf" |
"upd x y (Node2 l ab r) = (case cmp x (fst ab) of
LT ⇒ (case upd x y l of
T⇩i l' => T⇩i (Node2 l' ab r)
| Up⇩i l1 ab' l2 => T⇩i (Node3 l1 ab' l2 ab r)) |
EQ ⇒ T⇩i (Node2 l (x,y) r) |
GT ⇒ (case upd x y r of
T⇩i r' => T⇩i (Node2 l ab r')
| Up⇩i r1 ab' r2 => T⇩i (Node3 l ab r1 ab' r2)))" |
"upd x y (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
LT ⇒ (case upd x y l of
T⇩i l' => T⇩i (Node3 l' ab1 m ab2 r)
| Up⇩i l1 ab' l2 => Up⇩i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) |
EQ ⇒ T⇩i (Node3 l (x,y) m ab2 r) |
GT ⇒ (case cmp x (fst ab2) of
LT ⇒ (case upd x y m of
T⇩i m' => T⇩i (Node3 l ab1 m' ab2 r)
| Up⇩i m1 ab' m2 => Up⇩i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) |
EQ ⇒ T⇩i (Node3 l ab1 m (x,y) r) |
GT ⇒ (case upd x y r of
T⇩i r' => T⇩i (Node3 l ab1 m ab2 r')
| Up⇩i r1 ab' r2 => Up⇩i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))"

definition update :: "'a::linorder ⇒ 'b ⇒ ('a*'b) tree23 ⇒ ('a*'b) tree23" where
"update a b t = tree⇩i(upd a b t)"

fun del :: "'a::linorder ⇒ ('a*'b) tree23 ⇒ ('a*'b) up⇩d" where
"del x Leaf = T⇩d Leaf" |
"del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then Up⇩d Leaf else T⇩d(Node2 Leaf ab1 Leaf))" |
"del x (Node3 Leaf ab1 Leaf ab2 Leaf) = T⇩d(if x=fst ab1 then Node2 Leaf ab2 Leaf
else if x=fst ab2 then Node2 Leaf ab1 Leaf else Node3 Leaf ab1 Leaf ab2 Leaf)" |
"del x (Node2 l ab1 r) = (case cmp x (fst ab1) of
LT ⇒ node21 (del x l) ab1 r |
GT ⇒ node22 l ab1 (del x r) |
EQ ⇒ let (ab1',t) = split_min r in node22 l ab1' t)" |
"del x (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
LT ⇒ node31 (del x l) ab1 m ab2 r |
EQ ⇒ let (ab1',m') = split_min m in node32 l ab1' m' ab2 r |
GT ⇒ (case cmp x (fst ab2) of
LT ⇒ node32 l ab1 (del x m) ab2 r |
EQ ⇒ let (ab2',r') = split_min r in node33 l ab1 m ab2' r' |
GT ⇒ node33 l ab1 m ab2 (del x r)))"

definition delete :: "'a::linorder ⇒ ('a*'b) tree23 ⇒ ('a*'b) tree23" where
"delete x t = tree⇩d(del x t)"

subsection ‹Functional Correctness›

lemma lookup_map_of:
"sorted1(inorder t) ⟹ 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) ⟹ inorder(tree⇩i(upd x y t)) = upd_list x y (inorder t)"
by(induction t) (auto simp: upd_list_simps split: up⇩i.splits)

corollary inorder_update:
"sorted1(inorder t) ⟹ inorder(update x y t) = upd_list x y (inorder t)"

lemma inorder_del: "⟦ bal t ; sorted1(inorder t) ⟧ ⟹
inorder(tree⇩d (del x t)) = del_list x (inorder t)"
by(induction t rule: del.induct)
(auto simp: del_list_simps inorder_nodes split_minD split!: if_split prod.splits)

corollary inorder_delete: "⟦ bal t ; sorted1(inorder t) ⟧ ⟹
inorder(delete x t) = del_list x (inorder t)"

subsection ‹Balancedness›

lemma bal_upd: "bal t ⟹ bal (tree⇩i(upd x y t)) ∧ height(upd x y t) = height t"
by (induct t) (auto split!: if_split up⇩i.split)(* 16 secs in 2015 *)

corollary bal_update: "bal t ⟹ bal (update x y t)"

lemma height_del: "bal t ⟹ height(del x t) = height t"
by(induction x t rule: del.induct)
(auto simp add: heights max_def height_split_min split: prod.split)

lemma bal_tree⇩d_del: "bal t ⟹ bal(tree⇩d(del x t))"
by(induction x t rule: del.induct)
(auto simp: bals bal_split_min height_del height_split_min split: prod.split)

corollary bal_delete: "bal t ⟹ bal(delete x t)"

subsection ‹Overall Correctness›

interpretation M: Map_by_Ordered
where empty = empty and lookup = lookup and update = update and delete = delete
and inorder = inorder and inv = bal
proof (standard, goal_cases)
case 1 thus ?case by(simp add: empty_def)
next
case 2 thus ?case by(simp add: lookup_map_of)
next
case 3 thus ?case by(simp add: inorder_update)
next
case 4 thus ?case by(simp add: inorder_delete)
next
case 5 thus ?case by(simp add: empty_def)
next
case 6 thus ?case by(simp add: bal_update)
next
case 7 thus ?case by(simp add: bal_delete)
qed

end
```