65 lines
2.5 KiB
Text
65 lines
2.5 KiB
Text
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set_option structure.eta_thm true
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set_option structure.proj_mk_thm true
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structure semigroup [class] (A : Type) extends has_mul A :=
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(mul_assoc : ∀a b c, mul (mul a b) c = mul a (mul b c))
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structure comm_semigroup [class] (A : Type) extends semigroup A :=
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(mul_comm : ∀a b, mul a b = mul b a)
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structure left_cancel_semigroup [class] (A : Type) extends semigroup A :=
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(mul_left_cancel : ∀a b c, mul a b = mul a c → b = c)
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structure right_cancel_semigroup [class] (A : Type) extends semigroup A :=
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(mul_right_cancel : ∀a b c, mul a b = mul c b → a = c)
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structure add_semigroup [class] (A : Type) extends has_add A :=
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(add_assoc : ∀a b c, add (add a b) c = add a (add b c))
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structure add_comm_semigroup [class] (A : Type) extends add_semigroup A :=
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(add_comm : ∀a b, add a b = add b a)
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structure add_left_cancel_semigroup [class] (A : Type) extends add_semigroup A :=
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(add_left_cancel : ∀a b c, add a b = add a c → b = c)
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structure add_right_cancel_semigroup [class] (A : Type) extends add_semigroup A :=
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(add_right_cancel : ∀a b c, add a b = add c b → a = c)
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structure monoid [class] (A : Type) extends semigroup A, has_one A :=
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(one_mul : ∀a, mul one a = a) (mul_one : ∀a, mul a one = a)
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structure comm_monoid [class] (A : Type) extends monoid A, comm_semigroup A
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structure add_monoid [class] (A : Type) extends add_semigroup A, has_zero A :=
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(zero_add : ∀a, add zero a = a) (add_zero : ∀a, add a zero = a)
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structure add_comm_monoid [class] (A : Type) extends add_monoid A, add_comm_semigroup A
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structure group [class] (A : Type) extends monoid A, has_inv A :=
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(mul_left_inv : ∀a, mul (inv a) a = one)
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structure comm_group [class] (A : Type) extends group A, comm_monoid A
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structure add_group [class] (A : Type) extends add_monoid A, has_neg A :=
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(add_left_inv : ∀a, add (neg a) a = zero)
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structure add_comm_group [class] (A : Type) extends add_group A, add_comm_monoid A
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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 : ∀a, mul zero a = zero)
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(mul_zero : ∀a, 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, distrib A,
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mul_zero_class A, zero_ne_one_class A
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set_option pp.implicit true
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check @semiring.mul.mk
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check @semiring.eta
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