44 lines
1.6 KiB
Text
44 lines
1.6 KiB
Text
/-
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Copyright (c) 2016 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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prelude
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import init.logic
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universe variables u v
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variable {A : Type u}
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variable {B : Type v}
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variable f : A → A → A
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variable inv : A → A
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variable one : A
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local notation a * b := f a b
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local notation a ⁻¹ := inv a
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variable g : A → A → A
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local notation a + b := g a b
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def commutative := ∀ a b, a * b = b * a
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def associative := ∀ a b c, (a * b) * c = a * (b * c)
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def left_identity := ∀ a, one * a = a
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def right_identity := ∀ a, a * one = a
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def right_inverse := ∀ a, a * a⁻¹ = one
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def left_cancelative := ∀ a b c, a * b = a * c → b = c
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def right_cancelative := ∀ a b c, a * b = c * b → a = c
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def left_distributive := ∀ a b c, a * (b + c) = a * b + a * c
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def right_distributive := ∀ a b c, (a + b) * c = a * c + b * c
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def right_commutative (h : B → A → B) := ∀ b a₁ a₂, h (h b a₁) a₂ = h (h b a₂) a₁
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def left_commutative (h : A → B → B) := ∀ a₁ a₂ b, h a₁ (h a₂ b) = h a₂ (h a₁ b)
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lemma left_comm : commutative f → associative f → left_commutative f :=
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assume hcomm hassoc, take a b c, calc
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a*(b*c) = (a*b)*c : eq.symm (hassoc a b c)
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... = (b*a)*c : hcomm a b ▸ rfl
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... = b*(a*c) : hassoc b a c
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lemma right_comm : commutative f → associative f → right_commutative f :=
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assume hcomm hassoc, take a b c, calc
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(a*b)*c = a*(b*c) : hassoc a b c
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... = a*(c*b) : hcomm b c ▸ rfl
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... = (a*c)*b : eq.symm (hassoc a c b)
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