lean2/tests/lean/run/tree.lean

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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