src/HOL/Data_Structures/RBT_Map.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/RBT_Map.thy	Tue Sep 22 08:38:25 2015 +0200
@@ -0,0 +1,79 @@
+(* Author: Tobias Nipkow *)
+
+section \<open>Red-Black Tree Implementation of Maps\<close>
+
+theory RBT_Map
+imports
+  RBT_Set
+  Map_by_Ordered
+begin
+
+fun lookup :: "('a::linorder * 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b option" where
+"lookup Leaf x = None" |
+"lookup (Node _ l (a,b) r) x =
+  (if x < a then lookup l x else
+   if x > a then lookup r x else Some b)"
+
+fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
+"update x y Leaf = R Leaf (x,y) Leaf" |
+"update x y (B l (a,b) r) =
+  (if x < a then bal (update x y l) (a,b) r else
+   if x > a then bal l (a,b) (update x y r)
+   else B l (x,y) r)" |
+"update x y (R l (a,b) r) =
+  (if x < a then R (update x y l) (a,b) r else
+   if x > a then R l (a,b) (update x y r)
+   else R l (x,y) r)"
+
+fun delete :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
+and deleteL :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
+and deleteR :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
+where
+"delete x Leaf = Leaf" |
+"delete x (Node c t1 (a,b) t2) = 
+  (if x < a then deleteL x t1 (a,b) t2 else
+   if x > a then deleteR x t1 (a,b) t2 else combine t1 t2)" |
+"deleteL x (B t1 a t2) b t3 = balL (delete x (B t1 a t2)) b t3" |
+"deleteL x t1 a t2 = R (delete x t1) a t2" |
+"deleteR x t1 a (B t2 b t3) = balR t1 a (delete x (B t2 b t3))" | 
+"deleteR x t1 a t2 = R t1 a (delete x t2)"
+
+
+subsection "Functional Correctness Proofs"
+
+lemma lookup_eq:
+  "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
+by(induction t)
+  (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split)
+
+
+lemma inorder_update:
+  "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
+by(induction x y t rule: update.induct)
+  (auto simp: upd_list_simps inorder_bal)
+
+
+lemma inorder_delete:
+ "sorted1(inorder t1) \<Longrightarrow>  inorder(delete x t1) = del_list x (inorder t1)" and
+ "sorted1(inorder t1) \<Longrightarrow>  inorder(deleteL x t1 a t2) =
+    del_list x (inorder t1) @ a # inorder t2" and
+ "sorted1(inorder t2) \<Longrightarrow>  inorder(deleteR x t1 a t2) =
+    inorder t1 @ a # del_list x (inorder t2)"
+by(induction x t1 and x t1 a t2 and x t1 a t2 rule: delete_deleteL_deleteR.induct)
+  (auto simp: del_list_sorted sorted_lems inorder_combine inorder_balL inorder_balR)
+
+
+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