feat(library/theories/analysis/complex_norm): instantiate complex numbers as a real normed vector space
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5 changed files with 79 additions and 13 deletions
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@ -722,6 +722,11 @@ section
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apply zero_lt_one
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end
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lemma eq_zero_of_mul_self_add_mul_self_eq_zero {x y : A} (H : x * x + y * y = 0) : x = 0 :=
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have x * x ≤ (0 : A), from calc
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x * x ≤ x * x + y * y : le_add_of_nonneg_right (mul_self_nonneg y)
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... = 0 : H,
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eq_zero_of_mul_self_eq_zero (le.antisymm this (mul_self_nonneg x))
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end
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/- TODO: Multiplication and one, starting with mult_right_le_one_le. -/
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@ -185,16 +185,6 @@ has_sub.mk has_sub.sub
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theorem of_real_sub (x y : ℝ) : of_real (x - y) = of_real x - of_real y :=
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rfl
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-- TODO: move these
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private lemma eq_zero_of_mul_self_eq_zero {x : ℝ} (H : x * x = 0) : x = 0 :=
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iff.mp !or_self (!eq_zero_or_eq_zero_of_mul_eq_zero H)
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private lemma eq_zero_of_sum_square_eq_zero {x y : ℝ} (H : x * x + y * y = 0) : x = 0 :=
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have x * x ≤ (0 : ℝ), from calc
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x * x ≤ x * x + y * y : le_add_of_nonneg_right (mul_self_nonneg y)
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... = 0 : H,
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eq_zero_of_mul_self_eq_zero (le.antisymm this (mul_self_nonneg x))
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/- complex modulus and conjugate-/
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definition cmod (z : ℂ) : ℝ :=
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@ -208,8 +198,8 @@ by rewrite [↑cmod, re_of_real, im_of_real, mul_zero, add_zero]
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theorem eq_zero_of_cmod_eq_zero {z : ℂ} (H : cmod z = 0) : z = 0 :=
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have H1 : (complex.re z) * (complex.re z) + (complex.im z) * (complex.im z) = 0,
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from H,
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have H2 : complex.re z = 0, from eq_zero_of_sum_square_eq_zero H1,
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have H3 : complex.im z = 0, from eq_zero_of_sum_square_eq_zero (!add.comm ▸ H1),
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have H2 : complex.re z = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero H1,
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have H3 : complex.im z = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero (!add.comm ▸ H1),
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show z = 0, from complex.eq H2 H3
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definition conj (z : ℂ) : ℂ := complex.mk (complex.re z) (-(complex.im z))
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@ -7,3 +7,4 @@ theories.analysis
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* [ivt](ivt.lean) : the intermediate value theorem
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* [sqrt](sqrt.lean) : the sqrt function on the reals
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* [inner_product](inner_product.lean) : real inner product spaces
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* [complex_norm](complex_norm.lean) : the complex numbers as a real normed vector space
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70
library/theories/analysis/complex_norm.lean
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70
library/theories/analysis/complex_norm.lean
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@ -0,0 +1,70 @@
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/-
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Copyright (c) 2016 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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Instantiate the complex numbers as a normed space, by temporarily making it an inner product space
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over the reals.
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-/
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import theories.analysis.inner_product data.complex
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open nat real complex analysis classical
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noncomputable theory
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namespace complex
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namespace real_inner_product_space
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definition smul (a : ℝ) (z : ℂ) : ℂ := complex.mk (a * re z) (a * im z)
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definition ip (z w : ℂ) : ℝ := re z * re w + im z * im w
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proposition smul_left_distrib (a : ℝ) (z w : ℂ) : smul a (z + w) = smul a z + smul a w :=
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by rewrite [↑smul, *re_add, *im_add, *left_distrib]
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proposition smul_right_distrib (a b : ℝ) (z : ℂ) : smul (a + b) z = smul a z + smul b z :=
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by rewrite [↑smul, *right_distrib]
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proposition mul_smul (a b : ℝ) (z : ℂ) : smul (a * b) z = smul a (smul b z) :=
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by rewrite [↑smul, *mul.assoc]
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proposition one_smul (z : ℂ) : smul 1 z = z := by rewrite [↑smul, *one_mul, complex.eta]
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proposition inner_add_left (x y z : ℂ) : ip (x + y) z = ip x z + ip y z :=
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by rewrite [↑ip, re_add, im_add, *right_distrib, *add.assoc, add.left_comm (re y * re z)]
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proposition inner_smul_left (a : ℝ) (x y : ℂ) : ip (smul a x) y = a * ip x y :=
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by rewrite [↑ip, ↑smul, left_distrib, *mul.assoc]
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proposition inner_comm (x y : ℂ) : ip x y = ip y x :=
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by rewrite [↑ip, mul.comm, mul.comm (im x)]
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proposition inner_self_nonneg (x : ℂ) : ip x x ≥ 0 :=
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add_nonneg (mul_self_nonneg (re x)) (mul_self_nonneg (im x))
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proposition eq_zero_of_inner_self_eq_zero {x : ℂ} (H : ip x x = 0) : x = 0 :=
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have re x = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero H,
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have im x = 0, from eq_zero_of_mul_self_add_mul_self_eq_zero
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(by rewrite [↑ip at H, add.comm at H]; exact H),
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by+ rewrite [-complex.eta, `re x = 0`, `im x = 0`]
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end real_inner_product_space
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protected definition real_inner_product_space [reducible] : inner_product_space ℂ :=
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⦃ inner_product_space, complex.discrete_field,
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smul := real_inner_product_space.smul,
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inner := real_inner_product_space.ip,
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smul_left_distrib := real_inner_product_space.smul_left_distrib,
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smul_right_distrib := real_inner_product_space.smul_right_distrib,
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mul_smul := real_inner_product_space.mul_smul,
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one_smul := real_inner_product_space.one_smul,
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inner_add_left := real_inner_product_space.inner_add_left,
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inner_smul_left := real_inner_product_space.inner_smul_left,
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inner_comm := real_inner_product_space.inner_comm,
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inner_self_nonneg := real_inner_product_space.inner_self_nonneg,
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eq_zero_of_inner_self_eq_zero := @real_inner_product_space.eq_zero_of_inner_self_eq_zero
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⦄
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local attribute complex.real_inner_product_space [trans_instance]
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protected definition normed_vector_space [trans_instance] [reducible] : normed_vector_space ℂ :=
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_
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theorem norm_squared_eq_cmod (z : ℂ) : ∥ z ∥^2 = cmod z := by rewrite norm_squared
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end complex
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@ -1,5 +1,5 @@
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/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Copyright (c) 2016 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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