src/HOL/Data_Structures/Tree_Set.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Tree_Set.thy	Mon Sep 21 14:44:32 2015 +0200
@@ -0,0 +1,75 @@
+(* Author: Tobias Nipkow *)
+
+section {* Tree Implementation of Sets *}
+
+theory Tree_Set
+imports
+  "~~/src/HOL/Library/Tree"
+  Set_by_Ordered
+begin
+
+fun isin :: "'a::linorder tree \<Rightarrow> 'a \<Rightarrow> bool" where
+"isin Leaf x = False" |
+"isin (Node l a r) x = (x < a \<and> isin l x \<or> x=a \<or> isin r x)"
+
+hide_const (open) insert
+
+fun insert :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"insert a Leaf = Node Leaf a Leaf" |
+"insert a (Node l x r) =
+   (if a < x then Node (insert a l) x r
+    else if a=x then Node l x r
+    else Node l x (insert a 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 tree \<Rightarrow> 'a tree" where
+"delete k Leaf = Leaf" |
+"delete k (Node l a r) = (if k<a then Node (delete k l) a r else
+  if k > a then Node l a (delete k r) else
+  if r = Leaf then l else let (a',r') = del_min r in Node l a' r')"
+
+
+subsection "Functional Correctness Proofs"
+
+lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
+by (induction t) (auto simp: elems_simps)
+
+lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
+by (induction t) (auto simp: elems_simps0 dest: sortedD)
+
+
+lemma inorder_insert:
+  "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
+by(induction t) (auto simp: ins_simps)
+
+
+lemma del_minD:
+  "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted(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:
+  "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
+by(induction t) (auto simp: del_simps del_minD split: prod.splits)
+
+
+interpretation Set_by_Ordered
+where empty = Leaf and isin = isin and insert = insert 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: isin_set)
+next
+  case 3 thus ?case by(simp add: inorder_insert)
+next
+  case 4 thus ?case by(simp add: inorder_delete)
+next
+  case 5 thus ?case by(simp)
+qed
+
+end