21 lines
617 B
Text
21 lines
617 B
Text
class One (M : Type u) where one : M
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instance {M} [One M] : OfNat M (nat_lit 1) := ⟨One.one⟩
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class Monoid (M : Type u) extends Mul M, One M where
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mul_one (m : M) : m * 1 = m
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class AddCommMonoid (A : Type u) extends Add A, Zero A
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class MonoidWithZero (M₀ : Type u) extends Monoid M₀, Zero M₀
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class Semiring (R : Type u) extends AddCommMonoid R, MonoidWithZero R, One R
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#print Semiring -- only toMonoid field, no duplicate toOne
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def oneViaMonoid {M} [Monoid M] : M := 1
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example {R} [Semiring R] : (1 : R) = oneViaMonoid := rfl
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example : Semiring Nat where
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mul_one := by simp
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zero := 0
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one := 1
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