diff -r 947ce60a06e1 -r 00d9682e8dd7 src/HOL/Data_Structures/AVL_Map.thy
--- a/src/HOL/Data_Structures/AVL_Map.thy	Wed Nov 04 15:07:23 2015 +0100
+++ b/src/HOL/Data_Structures/AVL_Map.thy	Thu Nov 05 08:27:14 2015 +0100
@@ -8,36 +8,34 @@
   Lookup2
 begin
 
-fun update :: "'a::order \<Rightarrow> 'b \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
+fun update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
 "update x y Leaf = Node 1 Leaf (x,y) Leaf" |
-"update x y (Node h l (a,b) r) = 
-   (if x = a then Node h l (x,y) r else
-    if x < a then node_bal_l (update x y l) (a,b) r
-    else node_bal_r l (a,b) (update x y r))"
+"update x y (Node h l (a,b) r) = (case cmp x a of
+   EQ \<Rightarrow> Node h l (x,y) r |
+   LT \<Rightarrow> balL (update x y l) (a,b) r |
+   GT \<Rightarrow> balR l (a,b) (update x y r))"
 
-fun delete :: "'a::order \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
+fun delete :: "'a::cmp \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
 "delete _ Leaf = Leaf" |
-"delete x (Node h l (a,b) r) = (
-   if x = a then delete_root (Node h l (a,b) r) else
-   if x < a then node_bal_r (delete x l) (a,b) r
-   else node_bal_l l (a,b) (delete x r))"
+"delete x (Node h l (a,b) r) = (case cmp x a of
+   EQ \<Rightarrow> delete_root (Node h l (a,b) r) |
+   LT \<Rightarrow> balR (delete x l) (a,b) r |
+   GT \<Rightarrow> balL l (a,b) (delete x r))"
 
 
 subsection {* Functional Correctness Proofs *}
 
 theorem inorder_update:
   "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
-by (induct t) 
-   (auto simp: upd_list_simps inorder_node_bal_l inorder_node_bal_r)
+by (induct t) (auto simp: upd_list_simps inorder_balL inorder_balR)
 
 
 theorem inorder_delete:
   "sorted1(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
 by(induction t)
-  (auto simp: del_list_simps inorder_node_bal_l inorder_node_bal_r
+  (auto simp: del_list_simps inorder_balL inorder_balR
      inorder_delete_root inorder_delete_maxD split: prod.splits)
 
-
 interpretation Map_by_Ordered
 where empty = Leaf and lookup = lookup and update = update and delete = delete
 and inorder = inorder and wf = "\<lambda>_. True"