27 lines
677 B
Text
27 lines
677 B
Text
class Zero (α : Type u) where
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zero : α
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export Zero (zero)
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instance [Zero α] : OfNat α (nat_lit 0) where
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ofNat := zero
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class AddGroup (α : Type u) extends Add α, Zero α, Neg α where
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addAssoc : {a b c : α} → a + b + c = a + (b + c)
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zeroAdd : {a : α} → 0 + a = a
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addZero : {a : α} → a + 0 = a
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negAdd : {a : α} → -a + a = 0
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open AddGroup
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theorem negZero [AddGroup α] : -(0 : α) = 0 := by
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rw [←addZero (a := -(0 : α)), negAdd]
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theorem subZero [AddGroup α] {a : α} : a + -(0 : α) = a := by
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rw [← addZero (a := a)]
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rw addAssoc
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rw negZero
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rw addZero
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theorem shouldFail [AddGroup α] : ((0 : α) + 0) = 0 :=
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rfl -- Error
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