import logic data.prod open eq.ops prod inductive tree (A : Type) := leaf : A → tree A, node : tree A → tree A → tree A inductive one.{l} : Type.{max 1 l} := star : one set_option pp.universes true namespace tree namespace manual section universe variables l₁ l₂ variable {A : Type.{l₁}} variable (C : tree A → Type.{l₂}) definition below (t : tree A) : Type := rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂) end section universe variables l₁ l₂ variable {A : Type.{l₁}} variable {C : tree A → Type.{l₂}} definition below_rec_on (t : tree A) (H : Π (n : tree A), below C n → C n) : C t := have general : C t × below C t, from rec_on t (λa, (H (leaf a) one.star, one.star)) (λ (l r : tree A) (Hl : C l × below C l) (Hr : C r × below C r), have b : below C (node l r), from (pr₁ Hl, pr₁ Hr, pr₂ Hl, pr₂ Hr), have c : C (node l r), from H (node l r) b, (c, b)), pr₁ general end end manual section universe variables l₁ l₂ variable {A : Type.{l₁}} variable {C : tree A → Type.{l₂+1}} definition below_rec_on (t : tree A) (H : Π (n : tree A), @below A C n → C n) : C t := have general : C t × @below A C t, from rec_on t (λa, (H (leaf a) unit.star, unit.star)) (λ (l r : tree A) (Hl : C l × @below A C l) (Hr : C r × @below A C r), have b : @below A C (node l r), from ((pr₁ Hl, pr₂ Hl), (pr₁ Hr, pr₂ Hr)), have c : C (node l r), from H (node l r) b, (c, b)), pr₁ general end set_option pp.universes true theorem leaf_ne_tree {A : Type} (a : A) (l r : tree A) : leaf a ≠ node l r := assume h : leaf a = node l r, no_confusion h end tree