52 lines
1.6 KiB
Text
52 lines
1.6 KiB
Text
|
import data.prod data.unit
|
||
|
|
open prod
|
||
|
|
|
||
|
|
inductive tree (A : Type) : Type :=
|
||
|
|
node : A → forest A → tree A
|
||
|
|
with forest : Type :=
|
||
|
|
nil : forest A,
|
||
|
|
cons : tree A → forest A → forest A
|
||
|
|
|
||
|
|
namespace solution1
|
||
|
|
|
||
|
|
inductive tree_forest (A : Type) :=
|
||
|
|
of_tree : tree A → tree_forest A,
|
||
|
|
of_forest : forest A → tree_forest A
|
||
|
|
|
||
|
|
inductive same_kind {A : Type} : tree_forest A → tree_forest A → Type :=
|
||
|
|
is_tree : Π (t₁ t₂ : tree A), same_kind (tree_forest.of_tree t₁) (tree_forest.of_tree t₂),
|
||
|
|
is_forest : Π (f₁ f₂ : forest A), same_kind (tree_forest.of_forest f₁) (tree_forest.of_forest f₂)
|
||
|
|
|
||
|
|
definition to_tree {A : Type} (tf : tree_forest A) (t : tree A) : same_kind tf (tree_forest.of_tree t) → tree A :=
|
||
|
|
tree_forest.cases_on tf
|
||
|
|
(λ t₁ H, t₁)
|
||
|
|
(λ f₁ H, by cases H)
|
||
|
|
|
||
|
|
end solution1
|
||
|
|
|
||
|
|
namespace solution2
|
||
|
|
|
||
|
|
variables {A B : Type}
|
||
|
|
|
||
|
|
inductive same_kind : sum A B → sum A B → Prop :=
|
||
|
|
isl : Π (a₁ a₂ : A), same_kind (sum.inl B a₁) (sum.inl B a₂),
|
||
|
|
isr : Π (b₁ b₂ : B), same_kind (sum.inr A b₁) (sum.inr A b₂)
|
||
|
|
|
||
|
|
definition to_left (s : sum A B) (a : A) : same_kind s (sum.inl B a) → A :=
|
||
|
|
sum.cases_on s
|
||
|
|
(λ a₁ H, a₁)
|
||
|
|
(λ b₁ H, false.rec _ (by cases H))
|
||
|
|
|
||
|
|
definition to_right (s : sum A B) (b : B) : same_kind s (sum.inr A b) → B :=
|
||
|
|
sum.cases_on s
|
||
|
|
(λ a₁ H, false.rec _ (by cases H))
|
||
|
|
(λ b₁ H, b₁)
|
||
|
|
|
||
|
|
theorem to_left_inl (a₁ a₂ : A) (H : same_kind (sum.inl B a₁) (sum.inl B a₂)) : to_left (sum.inl B a₁) a₂ H = a₁ :=
|
||
|
|
rfl
|
||
|
|
|
||
|
|
theorem to_right_inr (b₁ b₂ : B) (H : same_kind (sum.inr A b₁) (sum.inr A b₂)) : to_right (sum.inr A b₁) b₂ H = b₁ :=
|
||
|
|
rfl
|
||
|
|
|
||
|
|
end solution2
|