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