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