src/HOL/Data_Structures/AVL_Map.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/AVL_Map.thy	Wed Sep 23 09:47:04 2015 +0200
@@ -0,0 +1,54 @@
+(* Author: Tobias Nipkow *)
+
+section "AVL Tree Implementation of Maps"
+
+theory AVL_Map
+imports
+  AVL_Set
+  Lookup2
+begin
+
+fun update :: "'a::order \<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))"
+
+fun delete :: "'a::order \<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))"
+
+
+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)
+
+
+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
+     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"
+proof (standard, goal_cases)
+  case 1 show ?case by simp
+next
+  case 2 thus ?case by(simp add: lookup_eq)
+next
+  case 3 thus ?case by(simp add: inorder_update)
+next
+  case 4 thus ?case by(simp add: inorder_delete)
+qed (rule TrueI)+
+
+end