src/HOL/Data_Structures/Tree_Set.thy

   (case cmp x a of
      LT \<Rightarrow> Node (insert x l) a r |
      EQ \<Rightarrow> Node l a r |
      GT \<Rightarrow> Node l a (insert x r))"
 
+text \<open>Deletion by replacing:\<close>
+
 fun split_min :: "'a tree \<Rightarrow> '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 \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where

 "delete x (Node l a r) =
   (case cmp x a of
      LT \<Rightarrow>  Node (delete x l) a r |
      GT \<Rightarrow>  Node l a (delete x r) |
      EQ \<Rightarrow> if r = Leaf then l else let (a',r') = split_min r in Node l a' r')"
+
+text \<open>Deletion by appending:\<close>
+
+fun app :: "('a::linorder)tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"app t Leaf = t" |
+"app Leaf t = t" |
+"app (Node t1 a t2) (Node t3 b t4) =
+  (case app t2 t3 of
+     Leaf \<Rightarrow> Node t1 a (Node Leaf b t4) |
+     Node u2 x u3 \<Rightarrow> Node (Node t1 a u2) x (Node u3 b t4))"
+
+fun delete2 :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"delete2 x Leaf = Leaf" |
+"delete2 x (Node l a r) =
+  (case cmp x a of
+     LT \<Rightarrow>  Node (delete2 x l) a r |
+     GT \<Rightarrow>  Node l a (delete2 x r) |
+     EQ \<Rightarrow> app l r)"
 
 
 subsection "Functional Correctness Proofs"
 
 lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"

   case 3 thus ?case by(simp add: inorder_insert)
 next
   case 4 thus ?case by(simp add: inorder_delete)
 qed (rule TrueI)+
 
+lemma inorder_app:
+  "inorder(app l r) = inorder l @ inorder r"
+by(induction l r rule: app.induct) (auto split: tree.split)
+
+lemma inorder_delete2:
+  "sorted(inorder t) \<Longrightarrow> inorder(delete2 x t) = del_list x (inorder t)"
+by(induction t) (auto simp: inorder_app del_list_simps)
+
+interpretation S2: Set_by_Ordered
+where empty = empty and isin = isin and insert = insert and delete = delete2
+and inorder = inorder and inv = "\<lambda>_. True"
+proof (standard, goal_cases)
+  case 1 show ?case by (simp add: empty_def)
+next
+  case 2 thus ?case by(simp add: isin_set)
+next
+  case 3 thus ?case by(simp add: inorder_insert)
+next
+  case 4 thus ?case by(simp add: inorder_delete2)
+qed (rule TrueI)+
+
 end