136 lines
4.8 KiB
Text
136 lines
4.8 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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-/
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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 + 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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/- linear maps -/
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structure is_linear_map [class] (R : Type) {M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) :=
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(additive : ∀ u v : M₁, T (u + v) = T u + T v)
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(homogeneous : ∀ a : R, ∀ u : M₁, T (a • u) = a • T u)
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proposition linear_map_additive (R : Type) {M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) [linT : is_linear_map R T] (u v : M₁) :
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T (u + v) = T u + T v :=
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is_linear_map.additive smul₁ smul₂ _ _ T u v
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proposition linear_map_homogeneous {R M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) [linT : is_linear_map R T] (a : R) (u : M₁) :
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T (a • u) = a • T u :=
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is_linear_map.homogeneous smul₁ smul₂ _ _ T a u
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proposition is_linear_map_id [instance] (R : Type) {M : Type}
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[smulRM : has_scalar R M] [has_addM : has_add M] :
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is_linear_map R (id : M → M) :=
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is_linear_map.mk (take u v, rfl) (take a u, rfl)
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section is_linear_map
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variables {R M₁ M₂ : Type}
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variable [ringR : ring R]
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variable [moduleRM₁ : left_module R M₁]
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variable [moduleRM₂ : left_module R M₂]
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include ringR moduleRM₁ moduleRM₂
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variable T : M₁ → M₂
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variable [is_linear_mapT : is_linear_map R T]
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include is_linear_mapT
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proposition linear_map_zero : T 0 = 0 :=
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calc
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T 0 = T ((0 : R) • 0) : zero_smul
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... = (0 : R) • T 0 : linear_map_homogeneous T
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... = 0 : zero_smul
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proposition linear_map_neg (u : M₁) : T (-u) = -(T u) :=
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by rewrite [-neg_one_smul, linear_map_homogeneous T, neg_one_smul]
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proposition linear_map_smul_add_smul (a b : R) (u v : M₁) :
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T (a • u + b • v) = a • T u + b • T v :=
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by rewrite [linear_map_additive R T, *linear_map_homogeneous T]
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end is_linear_map
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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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/- an example -/
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section
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variables (F V : Type)
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variables [field F] [vector_spaceFV : vector_space F V]
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variable T : V → V
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variable [is_linear_map F T]
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include vector_spaceFV
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example (a b : F) (u v : V) : T (a • u + b • v) = a • T u + b • T v :=
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!linear_map_smul_add_smul
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end
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