import logic data.nat.basic
open nat

inductive vector (A : Type) : nat → Type :=
vnil  : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)

namespace vector
  definition no_confusion2 {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → vector.no_confusion_type P v₁ v₂ :=
  assume H₁₂ : v₁ = v₂,
    begin
       show vector.no_confusion_type P v₁ v₂, from
       have aux : v₁ = v₁ → vector.no_confusion_type P v₁ v₁, from
        take H₁₁,
          begin
            apply (vector.cases_on v₁),
              exact (assume h : P, h),

              intros (n, a, v, h),
              apply (h rfl),
              repeat (apply rfl),
              repeat (apply heq.refl)
          end,
        eq.rec_on H₁₂ aux H₁₂
    end

  theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
  begin
     intro h, apply (vector.no_confusion h), intros, assumption
  end

  theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ == v₂ :=
  begin
    intro h, apply (vector.no_confusion h), intros, eassumption
  end
end vector