lean2/library/algebra/binary.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import logic.eq
open eq.ops
namespace binary
context
parameter {A : Type}
parameter f : A → A → A
infixl `*`:75 := f
definition commutative := ∀{a b}, a*b = b*a
definition associative := ∀{a b c}, (a*b)*c = a*(b*c)
end
context
parameter {A : Type}
parameter {f : A → A → A}
hypothesis H_comm : commutative f
hypothesis H_assoc : associative f
infixl `*`:75 := f
theorem left_comm : ∀a b c, a*(b*c) = b*(a*c) :=
take a b c, calc
a*(b*c) = (a*b)*c : H_assoc⁻¹
... = (b*a)*c : {H_comm}
... = b*(a*c) : H_assoc
theorem right_comm : ∀a b c, (a*b)*c = (a*c)*b :=
take a b c, calc
(a*b)*c = a*(b*c) : H_assoc
... = a*(c*b) : {H_comm}
... = (a*c)*b : H_assoc⁻¹
end
end binary