62 lines
1.9 KiB
Text
62 lines
1.9 KiB
Text
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 :=
|
||
tree.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
|
||
tree.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), @tree.below A C n → C n) : C t
|
||
:= have general : C t × @tree.below A C t, from
|
||
tree.rec_on t
|
||
(λa, (H (leaf a) unit.star, unit.star))
|
||
(λ (l r : tree A) (Hl : C l × @tree.below A C l) (Hr : C r × @tree.below A C r),
|
||
have b : @tree.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,
|
||
tree.no_confusion h
|
||
end tree
|