src/HOL/Library/Tree.thy

   where
     "left Leaf = Leaf"
   | "right Leaf = Leaf"
 
 lemma neq_Leaf_iff: "(t \<noteq> Leaf) = (\<exists>l a r. t = Node l a r)"
-by (cases t) auto
+  by (cases t) auto
 
 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)"
+  "subtrees Leaf = {Leaf}" |
+  "subtrees (Node l a r) = insert (Node l a r) (subtrees l \<union> subtrees r)"
 
 lemma set_treeE: "a \<in> set_tree t \<Longrightarrow> \<exists>l r. Node l a r \<in> subtrees t"
   by (induction t)(auto)
 
-lemma Node_notin_subtrees_if[simp]:
-  "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t"
+lemma Node_notin_subtrees_if[simp]: "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t"
   by (induction t) auto
 
-lemma in_set_tree_if:
-  "Node l a r \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
+lemma in_set_tree_if: "Node l a r \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
   by (metis Node_notin_subtrees_if)
 
 fun inorder :: "'a tree \<Rightarrow> 'a list" where
-"inorder Leaf = []" |
-"inorder (Node l x r) = inorder l @ [x] @ inorder r"
+  "inorder Leaf = []" |
+  "inorder (Node l x r) = inorder l @ [x] @ inorder r"
 
 lemma set_inorder[simp]: "set (inorder t) = set_tree t"
   by (induction t) auto
 
 subsection {* Binary Search Tree predicate *}
 
-inductive bst :: "'a::linorder tree \<Rightarrow> bool" where
-Leaf[intro!, simp]: "bst Leaf" |
-Node: "bst l \<Longrightarrow> bst r \<Longrightarrow> (\<And>x. x \<in> set_tree l \<Longrightarrow> x < a) \<Longrightarrow> (\<And>x. x \<in> set_tree r \<Longrightarrow> a < x) \<Longrightarrow>
-    bst (Node l a r)"
+fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where
+  "bst Leaf \<longleftrightarrow> True" |
+  "bst (Node l a r) \<longleftrightarrow> bst l \<and> bst r \<and> (\<forall>x\<in>set_tree l. x < a) \<and> (\<forall>x\<in>set_tree r. a < x)"
 
-lemma bst_imp_sorted: "bst t \<Longrightarrow> sorted (inorder t)"
-  by (induction rule: bst.induct) (auto simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
+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)
 
 end