82 lines
3 KiB
Text
82 lines
3 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura, Jeremy Avigad
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General properties of binary operations.
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-/
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import algebra.function
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open eq.ops function
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namespace binary
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section
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variable {A : Type}
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variables (op₁ : A → A → A) (inv : A → A) (one : A)
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local notation a * b := op₁ a b
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local notation a ⁻¹ := inv a
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local notation 1 := one
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definition commutative [reducible] := ∀a b, a * b = b * a
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definition associative [reducible] := ∀a b c, (a * b) * c = a * (b * c)
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definition left_identity [reducible] := ∀a, 1 * a = a
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definition right_identity [reducible] := ∀a, a * 1 = a
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definition left_inverse [reducible] := ∀a, a⁻¹ * a = 1
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definition right_inverse [reducible] := ∀a, a * a⁻¹ = 1
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definition left_cancelative [reducible] := ∀a b c, a * b = a * c → b = c
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definition right_cancelative [reducible] := ∀a b c, a * b = c * b → a = c
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definition inv_op_cancel_left [reducible] := ∀a b, a⁻¹ * (a * b) = b
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definition op_inv_cancel_left [reducible] := ∀a b, a * (a⁻¹ * b) = b
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definition inv_op_cancel_right [reducible] := ∀a b, a * b⁻¹ * b = a
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definition op_inv_cancel_right [reducible] := ∀a b, a * b * b⁻¹ = a
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variable (op₂ : A → A → A)
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local notation a + b := op₂ a b
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definition left_distributive [reducible] := ∀a b c, a * (b + c) = a * b + a * c
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definition right_distributive [reducible] := ∀a b c, (a + b) * c = a * c + b * c
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definition right_commutative [reducible] {B : Type} (f : B → A → B) := ∀ b a₁ a₂, f (f b a₁) a₂ = f (f b a₂) a₁
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definition left_commutative [reducible] {B : Type} (f : A → B → B) := ∀ a₁ a₂ b, f a₁ (f a₂ b) = f a₂ (f a₁ b)
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end
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section
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variable {A : Type}
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variable {f : A → A → A}
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variable H_comm : commutative f
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variable H_assoc : associative f
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local infixl `*` := f
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theorem left_comm : left_commutative f :=
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take a b c, calc
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a*(b*c) = (a*b)*c : H_assoc
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... = (b*a)*c : H_comm
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... = b*(a*c) : H_assoc
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theorem right_comm : right_commutative f :=
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take a b c, calc
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(a*b)*c = a*(b*c) : H_assoc
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... = a*(c*b) : H_comm
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... = (a*c)*b : H_assoc
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end
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section
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variable {A : Type}
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variable {f : A → A → A}
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variable H_assoc : associative f
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local infixl `*` := f
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theorem assoc4helper (a b c d) : (a*b)*(c*d) = a*((b*c)*d) :=
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calc
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(a*b)*(c*d) = a*(b*(c*d)) : H_assoc
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... = a*((b*c)*d) : H_assoc
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end
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definition right_commutative_compose_right [reducible]
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{A B : Type} (f : A → A → A) (g : B → A) (rcomm : right_commutative f) : right_commutative (compose_right f g) :=
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λ a b₁ b₂, !rcomm
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definition left_commutative_compose_left [reducible]
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{A B : Type} (f : A → A → A) (g : B → A) (lcomm : left_commutative f) : left_commutative (compose_left f g) :=
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λ a b₁ b₂, !lcomm
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end binary
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