lean2/library/standard/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
using eq_proofs
namespace binary
section
parameter {A : Type}
parameter f : A → A → A
infixl `*`:75 := f
abbreviation commutative := ∀a b, a*b = b*a
abbreviation associative := ∀a b c, (a*b)*c = a*(b*c)
end
section
parameter {A : Type}
parameter {f : A → A → A}
infixl `*`:75 := f
hypothesis H_comm : commutative f
hypothesis H_assoc : associative 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