Theory Tree_Set

theory Tree_Set
imports Tree Cmp Set_Specs
(* Author: Tobias Nipkow *)

section ‹Unbalanced Tree Implementation of Set›

theory Tree_Set

definition empty :: "'a tree" where
"empty == Leaf"

fun isin :: "'a::linorder 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)"

hide_const (open) insert

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

fun split_min :: "'a tree ⇒ 'a * 'a tree" where
"split_min (Node l a r) =
  (if l = Leaf then (a,r) else let (x,l') = split_min l in (x, Node l' a r))"

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

subsection "Functional Correctness Proofs"

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

lemma inorder_insert:
  "sorted(inorder t) ⟹ inorder(insert x t) = ins_list x (inorder t)"
by(induction t) (auto simp: ins_list_simps)

lemma split_minD:
  "split_min t = (x,t') ⟹ t ≠ Leaf ⟹ x # inorder t' = inorder t"
by(induction t arbitrary: t' rule: split_min.induct)
  (auto simp: sorted_lems split: prod.splits if_splits)

lemma inorder_delete:
  "sorted(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 S: Set_by_Ordered
where empty = empty and isin = isin and insert = insert and delete = delete
and inorder = inorder and inv = "λ_. True"
proof (standard, goal_cases)
  case 1 show ?case by (simp add: empty_def)
  case 2 thus ?case by(simp add: isin_set)
  case 3 thus ?case by(simp add: inorder_insert)
  case 4 thus ?case by(simp add: inorder_delete)
qed (rule TrueI)+