src/HOL/Library/Tree.thy

--- a/src/HOL/Library/Tree.thy	Fri Jul 25 17:13:30 2014 +0200
+++ b/src/HOL/Library/Tree.thy	Fri Jul 25 18:41:53 2014 +0200
@@ -18,6 +18,12 @@
 lemma set_tree_Node2: "set_tree(Node l x r) = insert x (set_tree l \<union> set_tree r)"
 by auto
 
+lemma finite_set_tree[simp]: "finite(set_tree t)"
+by(induction t) auto
+
+
+subsection "The set of subtrees"
+
 fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where
   "subtrees Leaf = {Leaf}" |
   "subtrees (Node l a r) = insert (Node l a r) (subtrees l \<union> subtrees r)"
@@ -31,6 +37,9 @@
 lemma in_set_tree_if: "Node l a r \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
   by (metis Node_notin_subtrees_if)
 
+
+subsection "Inorder list of entries"
+
 fun inorder :: "'a tree \<Rightarrow> 'a list" where
   "inorder Leaf = []" |
   "inorder (Node l x r) = inorder l @ [x] @ inorder r"
@@ -38,6 +47,7 @@
 lemma set_inorder[simp]: "set (inorder t) = set_tree t"
   by (induction t) auto
 
+
 subsection {* Binary Search Tree predicate *}
 
 fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where
@@ -47,4 +57,53 @@
 lemma (in linorder) bst_imp_sorted: "bst t \<Longrightarrow> sorted (inorder t)"
   by (induction t) (auto simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
 
+
+subsection "Deletion of the rightmost entry"
+
+fun del_rightmost :: "'a tree \<Rightarrow> 'a tree * 'a" where
+"del_rightmost (Node l a Leaf) = (l,a)" |
+"del_rightmost (Node l a r) = (let (r',x) = del_rightmost r in (Node l a r', x))"
+
+lemma del_rightmost_set_tree_if_bst:
+  "\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> Leaf \<rbrakk>
+  \<Longrightarrow> x \<in> set_tree t \<and> set_tree t' = set_tree t - {x}"
+apply(induction t arbitrary: t' rule: del_rightmost.induct)
+  apply (fastforce simp: ball_Un split: prod.splits)+
+done
+
+lemma del_rightmost_set_tree:
+  "\<lbrakk> del_rightmost t = (t',x);  t \<noteq> Leaf \<rbrakk> \<Longrightarrow> set_tree t = insert x (set_tree t')"
+apply(induction t arbitrary: t' rule: del_rightmost.induct)
+by (auto split: prod.splits) auto
+
+lemma del_rightmost_bst:
+  "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> Leaf \<rbrakk> \<Longrightarrow> bst t'"
+proof(induction t arbitrary: t' rule: del_rightmost.induct)
+  case (2 l a rl b rr)
+  let ?r = "Node rl b rr"
+  from "2.prems"(1) obtain r' where 1: "del_rightmost ?r = (r',x)" and [simp]: "t' = Node l a r'"
+    by(simp split: prod.splits)
+  from "2.prems"(2) 1 del_rightmost_set_tree[OF 1] show ?case by(auto)(simp add: "2.IH")
+qed auto
+
+
+lemma del_rightmost_greater: "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> Leaf \<rbrakk>
+  \<Longrightarrow> \<forall>a\<in>set_tree t'. a < x"
+proof(induction t arbitrary: t' rule: del_rightmost.induct)
+  case (2 l a rl b rr)
+  from "2.prems"(1) obtain r'
+  where dm: "del_rightmost (Node rl b rr) = (r',x)" and [simp]: "t' = Node l a r'"
+    by(simp split: prod.splits)
+  show ?case using "2.prems"(2) "2.IH"[OF dm] del_rightmost_set_tree_if_bst[OF dm]
+    by (fastforce simp add: ball_Un)
+qed simp_all
+
+(* should be moved but metis not available in much of Main *)
+lemma Max_insert1: "\<lbrakk> finite A;  \<forall>a\<in>A. a \<le> x \<rbrakk> \<Longrightarrow> Max(insert x A) = x"
+by (metis Max_in Max_insert Max_singleton antisym max_def)
+
+lemma del_rightmost_Max:
+  "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> Leaf \<rbrakk> \<Longrightarrow> x = Max(set_tree t)"
+by (metis Max_insert1 del_rightmost_greater del_rightmost_set_tree finite_set_tree less_le_not_le)
+
 end