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