Theory HOL-Data_Structures.Isin2

(* Author: Tobias Nipkow *)

section ‹Function \textit{isin} for Tree2›

theory Isin2
imports
  Tree2
  Cmp
  Set_Specs
begin

fun isin :: "('a::linorder*'b) tree ⇒ 'a ⇒ bool" where
"isin Leaf x = False" |
"isin (Node l (a,_) r) x =
  (case cmp x a of
     LT ⇒ isin l x |
     EQ ⇒ True |
     GT ⇒ isin r x)"

lemma isin_set_inorder: "sorted(inorder t) ⟹ isin t x = (x ∈ set(inorder t))"
by (induction t rule: tree2_induct) (auto simp: isin_simps)

lemma isin_set_tree: "bst t ⟹ isin t x ⟷ x ∈ set_tree t"
by(induction t rule: tree2_induct) auto

end