(* Author: Tobias Nipkow *)
section {* Unbalanced Tree as Map *}
theory Tree_Map
imports
Tree_Set
Map_by_Ordered
begin
fun lookup :: "('a::linorder*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
"lookup Leaf x = None" |
"lookup (Node l (a,b) r) x = (if x < a then lookup l x else
if x > a then lookup r x else Some b)"
fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
"update a b Leaf = Node Leaf (a,b) Leaf" |
"update a b (Node l (x,y) r) =
(if a < x then Node (update a b l) (x,y) r
else if a=x then Node l (a,b) r
else Node l (x,y) (update a b r))"
fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
"delete k Leaf = Leaf" |
"delete k (Node l (a,b) r) = (if k<a then Node (delete k l) (a,b) r else
if k > a then Node l (a,b) (delete k r) else
if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')"
subsection "Functional Correctness Proofs"
lemma lookup_eq:
"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_update:
"sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
by(induction t) (auto simp: upd_list_simps)
lemma del_minD:
"del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
x # inorder t' = inorder t"
by(induction t arbitrary: t' rule: del_min.induct)
(auto simp: del_list_simps split: prod.splits)
lemma inorder_delete:
"sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
interpretation Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
and inorder = inorder and wf = "\<lambda>_. True"
proof (standard, goal_cases)
case 1 show ?case by simp
next
case 2 thus ?case by(simp add: lookup_eq)
next
case 3 thus ?case by(simp add: inorder_update)
next
case 4 thus ?case by(simp add: inorder_delete)
qed (rule TrueI)+
end