src/HOL/Data_Structures/Tree_Map.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Tree_Map.thy	Mon Sep 21 14:44:32 2015 +0200
@@ -0,0 +1,72 @@
+(* Author: Tobias Nipkow *)
+
+section {* Unbalanced Tree as Map *}
+
+theory Tree_Map
+imports
+  "~~/src/HOL/Library/Tree"
+  Map_by_Ordered
+begin
+
+fun lookup :: "('a::linorder*'b) tree \<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) tree \<Rightarrow> ('a*'b) tree" where
+"update a b Leaf = Node Leaf (a,b) Leaf" |
+"update a b (Node l (x,y) r) =
+   (if a < x then Node (update a b l) (x,y) r
+    else if a=x then Node l (a,b) r
+    else Node l (x,y) (update a b r))"
+
+fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
+"del_min (Node Leaf a r) = (a, r)" |
+"del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))"
+
+fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
+"delete k Leaf = Leaf" |
+"delete k (Node l (a,b) r) = (if k<a then Node (delete k l) (a,b) r else
+  if k > a then Node l (a,b) (delete k r) else
+  if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')"
+
+
+subsection "Functional Correctness Proofs"
+
+lemma lookup_eq: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
+apply (induction t)
+apply (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split)
+done
+
+
+lemma inorder_update:
+  "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
+by(induction t) (auto simp: upd_list_sorteds sorted_lems)
+
+
+lemma del_minD:
+  "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
+   x # inorder t' = inorder t"
+by(induction t arbitrary: t' rule: del_min.induct)
+  (auto simp: sorted_lems split: prod.splits)
+
+lemma inorder_delete:
+  "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
+by(induction t)
+  (auto simp: del_list_sorted sorted_lems dest!: del_minD 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