21 lines
644 B
Text
21 lines
644 B
Text
import data.nat.basic
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open nat
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definition associative {A : Type} (op : A → A → A) := ∀a b c, op (op a b) c = op a (op b c)
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structure semigroup [class] (A : Type) :=
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mk {} :: (mul: A → A → A) (mul_assoc : associative mul)
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definition nat_semigroup [instance] : semigroup nat :=
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semigroup.mk nat.mul nat.mul.assoc
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example (a b c : nat) : (a * b) * c = a * (b * c) :=
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semigroup.mul_assoc a b c
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structure semigroup2 (A : Type) :=
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mk () :: (mul: A → A → A) (mul_assoc : associative mul)
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definition s := semigroup2.mk nat nat.mul nat.mul.assoc
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example (a b c : nat) : (a * b) * c = a * (b * c) :=
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semigroup2.mul_assoc nat s a b c
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