36 lines
1.1 KiB
Text
36 lines
1.1 KiB
Text
import logic data.nat
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open nat algebra
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inductive tree (A : Type) :=
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| leaf : A → tree A
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| node : tree A → tree A → tree A
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namespace tree
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definition height {A : Type} (t : tree A) : nat :=
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tree.rec_on t
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(λ a, zero)
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(λ t₁ t₂ h₁ h₂, succ (max h₁ h₂))
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definition height_lt {A : Type} : tree A → tree A → Prop :=
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inv_image lt (@height A)
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definition height_lt.wf (A : Type) : well_founded (@height_lt A) :=
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inv_image.wf height lt.wf
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theorem height_lt.node_left {A : Type} (t₁ t₂ : tree A) : height_lt t₁ (node t₁ t₂) :=
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lt_succ_of_le (le_max_left (height t₁) (height t₂))
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theorem height_lt.node_right {A : Type} (t₁ t₂ : tree A) : height_lt t₂ (node t₁ t₂) :=
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lt_succ_of_le (le_max_right (height t₁) (height t₂))
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theorem height_lt.trans {A : Type} : transitive (@height_lt A) :=
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inv_image.trans lt height @nat.lt_trans
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example : height_lt (leaf (2:nat)) (node (leaf 1) (leaf 2)) :=
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height_lt.node_right _ _
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example : height_lt (leaf (2:nat)) (node (node (leaf 1) (leaf 2)) (leaf 3)) :=
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height_lt.trans (height_lt.node_right _ _) (height_lt.node_left _ _)
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end tree
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