72 lines
2.6 KiB
Text
72 lines
2.6 KiB
Text
/-
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Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Nathaniel Thomas, Jeremy Avigad
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Modules and vector spaces over a ring.
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(We use "left_module," which is more precise, because "module" is a keyword.)
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-/
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import algebra.field
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structure has_scalar [class] (F V : Type) :=
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(smul : F → V → V)
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infixl ` • `:73 := has_scalar.smul
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/- modules over a ring -/
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structure left_module [class] (R M : Type) [ringR : ring R]
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extends has_scalar R M, add_comm_group M :=
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(smul_left_distrib : ∀ (r : R) (x y : M), smul r (add x y) = (add (smul r x) (smul r y)))
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(smul_right_distrib : ∀ (r s : R) (x : M), smul (ring.add r s) x = (add (smul r x) (smul s x)))
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(mul_smul : ∀ r s x, smul (mul r s) x = smul r (smul s x))
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(one_smul : ∀ x, smul one x = x)
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section left_module
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variables {R M : Type}
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variable [ringR : ring R]
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variable [moduleRM : left_module R M]
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include ringR moduleRM
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-- Note: the anonymous include does not work in the propositions below.
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proposition smul_left_distrib (a : R) (u v : M) : a • (u + v) = a • u + a • v :=
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!left_module.smul_left_distrib
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proposition smul_right_distrib (a b : R) (u : M) : (a + b) • u = a • u + b • u :=
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!left_module.smul_right_distrib
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proposition mul_smul (a : R) (b : R) (u : M) : (a * b) • u = a • (b • u) :=
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!left_module.mul_smul
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proposition one_smul (u : M) : (1 : R) • u = u := !left_module.one_smul
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proposition zero_smul (u : M) : (0 : R) • u = 0 :=
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have (0 : R) • u + 0 • u = 0 • u + 0, by rewrite [-smul_right_distrib, *add_zero],
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!add.left_cancel this
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proposition smul_zero (a : R) : a • (0 : M) = 0 :=
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have a • (0:M) + a • 0 = a • 0 + 0, by rewrite [-smul_left_distrib, *add_zero],
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!add.left_cancel this
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proposition neg_smul (a : R) (u : M) : (-a) • u = - (a • u) :=
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eq_neg_of_add_eq_zero (by rewrite [-smul_right_distrib, add.left_inv, zero_smul])
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proposition neg_one_smul (u : M) : -(1 : R) • u = -u :=
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by rewrite [neg_smul, one_smul]
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proposition smul_neg (a : R) (u : M) : a • (-u) = -(a • u) :=
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by rewrite [-neg_one_smul, -mul_smul, mul_neg_one_eq_neg, neg_smul]
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proposition smul_sub_left_distrib (a : R) (u v : M) : a • (u - v) = a • u - a • v :=
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by rewrite [sub_eq_add_neg, smul_left_distrib, smul_neg]
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proposition sub_smul_right_distrib (a b : R) (v : M) : (a - b) • v = a • v - b • v :=
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by rewrite [sub_eq_add_neg, smul_right_distrib, neg_smul]
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end left_module
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/- vector spaces -/
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structure vector_space [class] (F V : Type) [fieldF : field F]
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extends left_module F V
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