src/HOL/Data_Structures/Tree234_Map.thy

 lemma inorder_upd:
   "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)"
 by(induction t)
   (auto simp: upd_list_simps, auto simp: upd_list_simps split: up\<^sub>i.splits)
 
-lemma inorder_update:
+lemma inorder_update_234:
   "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
 by(simp add: update_def inorder_upd)
 
 lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
   inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
 by(induction t rule: del.induct)
   (auto simp: del_list_simps inorder_nodes split_minD split!: if_splits prod.splits)
 (* 30 secs (2016) *)
 
-lemma inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
+lemma inorder_delete_234: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
   inorder(delete x t) = del_list x (inorder t)"
 by(simp add: delete_def inorder_del)
 
 
 subsection \<open>Balancedness\<close>

 
 
 subsection \<open>Overall Correctness\<close>
 
 interpretation Map_by_Ordered
-where empty = Leaf and lookup = lookup and update = update and delete = delete
+where empty = empty and lookup = lookup and update = update and delete = delete
 and inorder = inorder and inv = bal
 proof (standard, goal_cases)
   case 2 thus ?case by(simp add: lookup_map_of)
 next
-  case 3 thus ?case by(simp add: inorder_update)
+  case 3 thus ?case by(simp add: inorder_update_234)
 next
-  case 4 thus ?case by(simp add: inorder_delete)
+  case 4 thus ?case by(simp add: inorder_delete_234)
 next
   case 6 thus ?case by(simp add: bal_update)
 next
   case 7 thus ?case by(simp add: bal_delete)
-qed simp+
+qed (simp add: empty_def)+
 
 end