24 lines
740 B
Text
24 lines
740 B
Text
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import types.sigma types.prod
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import algebra.binary algebra.group
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open eq eq.ops
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namespace path_algebra
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variable {A : Type}
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structure distrib [class] (A : Type) extends has_mul A, has_add A :=
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(left_distrib : ∀a b c, mul a (add b c) = add (mul a b) (mul a c))
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(right_distrib : ∀a b c, mul (add a b) c = add (mul a c) (mul b c))
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structure mul_zero_class [class] (A : Type) extends has_mul A, has_zero A :=
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(zero_mul : Πa, mul zero a = zero)
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(mul_zero : Πa, mul a zero = zero)
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structure zero_ne_one_class [class] (A : Type) extends has_zero A, has_one A :=
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(zero_ne_one : zero ≠ one)
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structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A,
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distrib A, mul_zero_class A, zero_ne_one_class A
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end path_algebra
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