30 lines
946 B
Text
30 lines
946 B
Text
universe variables u
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inductive Vec (α : Type u) : nat → Type (max 1 u)
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| nil : Vec 0
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| cons : ∀ {n}, α → Vec n → Vec (nat.succ n)
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lemma split {α : Type u} {n : nat} (v : Vec α n) : (v == (Vec.nil α) ∧ n = 0) ∨ ∃ m h (t : Vec α m), v == Vec.cons h t ∧ n = nat.succ m :=
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Vec.cases_on v
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(or.inl ⟨heq.refl _, rfl⟩)
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(λ n h t, or.inr ⟨n, h, t, heq.refl _, rfl⟩)
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constant f {α : Type u} {n : nat} : Vec α n → nat
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axiom fax1 (α : Type u) : f (Vec.nil α) = 0
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axiom fax2 {α : Type u} {n : nat} (v : Vec α (nat.succ n)) : f v = 1
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example {α : Type u} {n : nat} (v : Vec α n) : f v ≠ 2 :=
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begin
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destruct v,
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{intros, intro, have h := fax1 α, cc},
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intros n1 h t, intros, intro, have h := fax2 (Vec.cons h t), cc
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end
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open nat
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example : ∀ n, 0 < n → succ (pred n) = n :=
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begin
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intro n,
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destruct n,
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{dsimp, intros, have h := lt_irrefl 0, cc},
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{intros, subst n, dsimp, reflexivity}
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end
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