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refactor(algebra/binary): remove unnecessary annotations

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Leonardo de Moura 2016-02-25 15:11:52 -08:00
parent 768ba1c363
commit d501295ec1
1 changed files with 18 additions and 18 deletions
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36
hott/algebra/binary.hlean
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@ -15,29 +15,29 @@ namespace binary
local notation a * b := op₁ a b
local notation a ⁻¹ := inv a
definition commutative [reducible] := Πa b, a * b = b * a
definition associative [reducible] := Πa b c, (a * b) * c = a * (b * c)
definition left_identity [reducible] := Πa, one * a = a
definition right_identity [reducible] := Πa, a * one = a
definition left_inverse [reducible] := Πa, a⁻¹ * a = one
definition right_inverse [reducible] := Πa, a * a⁻¹ = one
definition left_cancelative [reducible] := Πa b c, a * b = a * c → b = c
definition right_cancelative [reducible] := Πa b c, a * b = c * b → a = c
definition commutative := Πa b, a * b = b * a
definition associative := Πa b c, (a * b) * c = a * (b * c)
definition left_identity := Πa, one * a = a
definition right_identity := Πa, a * one = a
definition left_inverse := Πa, a⁻¹ * a = one
definition right_inverse := Πa, a * a⁻¹ = one
definition left_cancelative := Πa b c, a * b = a * c → b = c
definition right_cancelative := Πa b c, a * b = c * b → a = c
definition inv_op_cancel_left [reducible] := Πa b, a⁻¹ * (a * b) = b
definition op_inv_cancel_left [reducible] := Πa b, a * (a⁻¹ * b) = b
definition inv_op_cancel_right [reducible] := Πa b, a * b⁻¹ * b = a
definition op_inv_cancel_right [reducible] := Πa b, a * b * b⁻¹ = a
definition inv_op_cancel_left := Πa b, a⁻¹ * (a * b) = b
definition op_inv_cancel_left := Πa b, a * (a⁻¹ * b) = b
definition inv_op_cancel_right := Πa b, a * b⁻¹ * b = a
definition op_inv_cancel_right := Πa b, a * b * b⁻¹ = a
variable (op₂ : A → A → A)
local notation a + b := op₂ a b
definition left_distributive [reducible] := Πa b c, a * (b + c) = a * b + a * c
definition right_distributive [reducible] := Πa b c, (a + b) * c = a * c + b * c
definition left_distributive := Πa b c, a * (b + c) = a * b + a * c
definition right_distributive := Πa b c, (a + b) * c = a * c + b * c
definition right_commutative [reducible] {B : Type} (f : B → A → B) := Π b a₁ a₂, f (f b a₁) a₂ = f (f b a₂) a₁
definition left_commutative [reducible] {B : Type} (f : A → B → B) := Π a₁ a₂ b, f a₁ (f a₂ b) = f a₂ (f a₁ b)
definition right_commutative {B : Type} (f : B → A → B) := Π b a₁ a₂, f (f b a₁) a₂ = f (f b a₂) a₁
definition left_commutative {B : Type} (f : A → B → B) := Π a₁ a₂ b, f a₁ (f a₂ b) = f a₂ (f a₁ b)
end
section
@ -76,11 +76,11 @@ namespace binary
... = a*((b*c)*d) : H_assoc
end
definition right_commutative_compose_right [reducible]
definition right_commutative_compose_right
{A B : Type} (f : A → A → A) (g : B → A) (rcomm : right_commutative f) : right_commutative (compose_right f g) :=
λ a b₁ b₂, !rcomm
definition left_commutative_compose_left [reducible]
definition left_commutative_compose_left
{A B : Type} (f : A → A → A) (g : B → A) (lcomm : left_commutative f) : left_commutative (compose_left f g) :=
λ a b₁ b₂, !lcomm
end binary