# Theory HOL-Data_Structures.Tree_Map

```(* Author: Tobias Nipkow *)

section ‹Unbalanced Tree Implementation of Map›

theory Tree_Map
imports
Tree_Set
Map_Specs
begin

fun lookup :: "('a::linorder*'b) tree ⇒ 'a ⇒ 'b option" where
"lookup Leaf x = None" |
"lookup (Node l (a,b) r) x =
(case cmp x a of LT ⇒ lookup l x | GT ⇒ lookup r x | EQ ⇒ Some b)"

fun update :: "'a::linorder ⇒ 'b ⇒ ('a*'b) tree ⇒ ('a*'b) tree" where
"update x y Leaf = Node Leaf (x,y) Leaf" |
"update x y (Node l (a,b) r) = (case cmp x a of
LT ⇒ Node (update x y l) (a,b) r |
EQ ⇒ Node l (x,y) r |
GT ⇒ Node l (a,b) (update x y r))"

fun delete :: "'a::linorder ⇒ ('a*'b) tree ⇒ ('a*'b) tree" where
"delete x Leaf = Leaf" |
"delete x (Node l (a,b) r) = (case cmp x a of
LT ⇒ Node (delete x l) (a,b) r |
GT ⇒ Node l (a,b) (delete x r) |
EQ ⇒ if r = Leaf then l else let (ab',r') = split_min r in Node l ab' r')"

subsection "Functional Correctness Proofs"

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_update:
"sorted1(inorder t) ⟹ inorder(update a b t) = upd_list a b (inorder t)"
by(induction t) (auto simp: upd_list_simps)

lemma inorder_delete:
"sorted1(inorder t) ⟹ inorder(delete x t) = del_list x (inorder t)"
by(induction t) (auto simp: del_list_simps split_minD split: prod.splits)

interpretation M: Map_by_Ordered
where empty = empty and lookup = lookup and update = update and delete = delete
and inorder = inorder and inv = "λ_. True"
proof (standard, goal_cases)
case 1 show ?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)
qed auto

end
```