46 lines
1.5 KiB
Text
46 lines
1.5 KiB
Text
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import logic data.nat.basic
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open nat
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inductive vector (A : Type) : nat → Type :=
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vnil : vector A zero,
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vcons : Π {n : nat}, A → vector A n → vector A (succ n)
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namespace vector
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definition no_confusion_type {A : Type} {n : nat} (P : Type) (v₁ v₂ : vector A n) : Type :=
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cases_on v₁
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(cases_on v₂
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(P → P)
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(λ n₂ h₂ t₂, P))
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(λ n₁ h₁ t₁, cases_on v₂
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P
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(λ n₂ h₂ t₂, (Π (H : n₁ = n₂), h₁ = h₂ → eq.rec_on H t₁ = t₂ → P) → P))
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definition no_confusion {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
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assume H₁₂ : v₁ = v₂,
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begin
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show no_confusion_type P v₁ v₂, from
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have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
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take H₁₁,
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begin
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apply (cases_on v₁),
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exact (assume h : P, h),
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intros (n, a, v, h),
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apply (h rfl),
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repeat (apply rfl)
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end,
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eq.rec_on H₁₂ aux H₁₂
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end
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theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
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begin
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intro h, apply (no_confusion h), intros, assumption
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end
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theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ = v₂ :=
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begin
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intro h, apply (no_confusion h), intros, eassumption
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end
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end vector
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