37 lines
982 B
Text
37 lines
982 B
Text
-- 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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-- Author: Leonardo de Moura
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import logic.connectives.eq
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using eq_ops
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namespace binary
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section
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parameter {A : Type}
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parameter f : A → A → A
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infixl `*`:75 := f
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abbreviation commutative := ∀a b, a*b = b*a
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abbreviation associative := ∀a b c, (a*b)*c = a*(b*c)
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end
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section
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parameter {A : Type}
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parameter {f : A → A → A}
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infixl `*`:75 := f
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hypothesis H_comm : commutative f
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hypothesis H_assoc : associative f
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theorem left_comm : ∀a b c, a*(b*c) = b*(a*c) :=
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take a b c, calc
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a*(b*c) = (a*b)*c : (H_assoc _ _ _)⁻¹
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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 : ∀a b c, (a*b)*c = (a*c)*b :=
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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 _ _ _)⁻¹
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end
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end binary
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