85 lines
2.7 KiB
Text
85 lines
2.7 KiB
Text
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import logic data.prod
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open eq.ops prod tactic
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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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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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definition no_confusion_type {A : Type} (P : Type) (t₁ t₂ : tree A) : Type :=
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cases_on t₁
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(λ a₁, cases_on t₂
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(λ a₂, (a₁ = a₂ → P) → P)
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(λ l₂ r₂, P))
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(λ l₁ r₁, cases_on t₂
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(λ a₂, P)
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(λ l₂ r₂, (l₁ = l₂ → r₁ = r₂ → P) → P))
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set_option pp.universes true
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check no_confusion_type
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definition no_confusion {A : Type} {P : Type} {t₁ t₂ : tree A} : t₁ = t₂ → no_confusion_type P t₁ t₂ :=
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assume e₁ : t₁ = t₂,
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have aux₁ : t₁ = t₁ → no_confusion_type P t₁ t₁, from
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take h, cases_on t₁
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(λ a, assume h : a = a → P, h (eq.refl a))
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(λ l r, assume h : l = l → r = r → P, h (eq.refl l) (eq.refl r)),
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eq.rec aux₁ e₁ e₁
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check no_confusion
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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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constant A : Type₁
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constants l₁ l₂ r₁ r₂ : tree A
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axiom node_eq : node l₁ r₁ = node l₂ r₂
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check no_confusion node_eq
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definition tst : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂ := no_confusion node_eq
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check tst (λ e₁ e₂, e₁)
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theorem node.inj1 {A : Type} (l₁ l₂ r₁ r₂ : tree A) : node l₁ r₁ = node l₂ r₂ → l₁ = l₂ :=
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assume h,
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have trivial : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂, from no_confusion h,
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trivial (λ e₁ e₂, e₁)
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theorem node.inj2 {A : Type} (l₁ l₂ r₁ r₂ : tree A) : node l₁ r₁ = node l₂ r₂ → l₁ = l₂ :=
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begin
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intro h,
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apply (no_confusion h),
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intros, assumption
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end
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end tree
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