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theory Tree2
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imports Main
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begin
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datatype ('a,'b) tree =
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Leaf ("\<langle>\<rangle>") |
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62160
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Node 'b "('a,'b)tree" 'a "('a,'b) tree" ("(1\<langle>_,/ _,/ _,/ _\<rangle>)")
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61224
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fun inorder :: "('a,'b)tree \<Rightarrow> 'a list" where
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"inorder Leaf = []" |
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"inorder (Node _ l a r) = inorder l @ a # inorder r"
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fun height :: "('a,'b) tree \<Rightarrow> nat" where
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"height Leaf = 0" |
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"height (Node _ l a r) = max (height l) (height r) + 1"
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67967
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fun set_tree :: "('a,'b) tree \<Rightarrow> 'a set" where
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"set_tree Leaf = {}" |
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"set_tree (Node _ l x r) = Set.insert x (set_tree l \<union> set_tree r)"
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fun bst :: "('a::linorder,'b) tree \<Rightarrow> bool" where
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"bst Leaf = True" |
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"bst (Node _ l a r) = (bst l \<and> bst r \<and> (\<forall>x \<in> set_tree l. x < a) \<and> (\<forall>x \<in> set_tree r. a < x))"
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62650
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definition size1 :: "('a,'b) tree \<Rightarrow> nat" where
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"size1 t = size t + 1"
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lemma size1_simps[simp]:
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"size1 \<langle>\<rangle> = 1"
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"size1 \<langle>u, l, x, r\<rangle> = size1 l + size1 r"
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by (simp_all add: size1_def)
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lemma size1_ge0[simp]: "0 < size1 t"
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by (simp add: size1_def)
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68411
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lemma finite_set_tree[simp]: "finite(set_tree t)"
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by(induction t) auto
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62390
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end
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