39 lines
1.3 KiB
Text
39 lines
1.3 KiB
Text
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import logic
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open eq.ops
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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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namespace tree
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definition cases_on {A : Type} {C : tree A → Type} (t : tree A) (e₁ : Πa, C (leaf a)) (e₂ : Πt₁ t₂, C (node t₁ t₂)) : C t :=
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rec e₁ (λt₁ t₂ r₁ r₂, e₂ t₁ t₂) t
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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 false (leaf a) (node l r) h
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end tree
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