(* Author: Tobias Nipkow *)
section \<open>Function \textit{lookup} for Tree2\<close>
theory Lookup2
imports
Tree2
Cmp
Map_Specs
begin
fun lookup :: "(('a::linorder * 'b) * 'c) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
"lookup Leaf x = None" |
"lookup (Node l ((a,b), _) r) x =
(case cmp x a of LT \<Rightarrow> lookup l x | GT \<Rightarrow> lookup r x | EQ \<Rightarrow> Some b)"
lemma lookup_map_of:
"sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
by(induction t rule: tree2_induct) (auto simp: map_of_simps split: option.split)
end