| {"id":"def:rotated_roots","label":"rotated_roots","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"When the \\(N\\)-th roots of unity are rotated by an angle \\(\\theta \\) (equivalently, multiplied by \\(\\mathrm{e}^{i\\theta }\\)) and projected onto the real axis, we obtain the \\(N\\) real numbers \\begin{equation} r_k(\\theta ) = \\cos \\left(\\theta + \\frac{2\\pi k}{N}\\right), \\quad k = 0, 1, \\ldots , N-1. \\end{equation}","doc_url":"../../docs/find/#doc/realProjection_eq_cos","latex_link":"sect0001.html#def:rotated_roots","lean_link":"../../docs/find/#doc/realProjection_eq_cos","lean_decl_name":"realProjection_eq_cos","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/RootsOfUnity.lean","lean4_code":"theorem realProjection_eq_cos (N : ℕ) (θ : ℝ) (k : ℕ) (_hk : k < N) :\n (exp (I * (θ + 2 * π * k / N))).re = Real.cos (θ + 2 * π * k / N) := by\n rw [mul_comm]\n convert exp_ofReal_mul_I_re (θ + 2 * π * k / N)\n norm_cast\n\n/-- All real projections lie in the closed interval [-1, 1]. -/\n","line":40,"end_line":46,"proof_status":"formalized"} |
| {"id":"def:unscaled_poly","label":"unscaled_poly","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["def:rotated_roots"],"nlp_text":"Let \\(P_N(x; \\theta )\\) denote the monic polynomial having the rotated roots as roots: \\begin{equation} P_N(x; \\theta ) = \\prod _{k=0}^{N-1} \\left(x - r_k(\\theta )\\right). \\end{equation}","doc_url":"../../docs/find/#doc/polynomialFromRealRoots_degree","latex_link":"sect0001.html#def:unscaled_poly","lean_link":"../../docs/find/#doc/polynomialFromRealRoots_degree","lean_decl_name":"polynomialFromRealRoots_degree","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PolynomialConstruction.lean","lean4_code":"theorem polynomialFromRealRoots_degree (roots : List ℝ) :\n (polynomialFromRealRoots roots).degree = roots.length := by\n induction roots with\n | nil => simp [polynomialFromRealRoots]\n | cons r rs ih =>\n unfold polynomialFromRealRoots at ih ⊢\n simp only [List.foldr, List.length_cons]\n rw [degree_mul, degree_X_sub_C, ih]\n norm_cast\n ring\n\n/-- The unscaled polynomial has degree N. -/\n","line":58,"end_line":69,"proof_status":"formalized"} |
| {"id":"def:scaled_poly","label":"scaled_poly","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["def:unscaled_poly"],"nlp_text":"To match the normalization of Chebyshev polynomials, which have leading coefficient \\(2^{N-1}\\) for \\(N \\geq 1\\), we define the scaled polynomial: \\begin{equation} S_N(x; \\theta ) = 2^{N-1} \\cdot P_N(x; \\theta ). \\end{equation}","doc_url":"../../docs/find/#doc/scaledPolynomial_degree","latex_link":"sect0001.html#def:scaled_poly","lean_link":"../../docs/find/#doc/scaledPolynomial_degree","lean_decl_name":"scaledPolynomial_degree","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PolynomialConstruction.lean","lean4_code":"theorem scaledPolynomial_degree (N : ℕ) (θ : ℝ) (_hN : 0 < N) :\n (scaledPolynomial N θ).degree = N := by\n unfold scaledPolynomial\n rw [degree_C_mul, unscaledPolynomial_degree N θ]\n exact pow_ne_zero (N - 1) two_ne_zero\n\n/-- The unscaled polynomial is monic (leading coefficient is 1). -/\n","line":76,"end_line":82,"proof_status":"formalized"} |
| {"id":"thm:main","label":"main","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["def:scaled_poly","lem:chebyshev_roots","thm:coefficients_independent"],"nlp_text":"For any positive integer \\(N \\geq 1\\) and any angle \\(\\theta \\in \\mathbb {R}\\), \"> \\begin{equation} \\label{eq:main_result} S_N(x; \\theta ) = T_N(x) + c(\\theta ), \\end{equation}","doc_url":"../../docs/find/#doc/mainTheorem","latex_link":"sect0001.html#thm:main","lean_link":"../../docs/find/#doc/mainTheorem","lean_decl_name":"mainTheorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./index.html","lean4_code":"theorem mainTheorem (N : ℕ) (θ : ℝ) (hN : 0 < N) :<br>\n scaledPolynomial N θ = Polynomial.Chebyshev.T ℝ N + C (explicitConstant N θ)\n </div>\n <p style=\"margin-top: 0.75rem; font-size: 0.9rem; color: #555;\">\n where <code>explicitConstant N θ = 2^(N-1) * (-1)^N * ∏_{k=0}^{N-1} cos(θ + 2πk/N) - T_N(0)</code>\n </p>\n <p style=\"margin-top: 1rem; font-size: 0.95rem;\">\n <a href=\"./docs/ChebyshevCircles/ProofOfMainTheorem.html#mainTheorem\" style=\"color: #3498db; text-decoration: none; font-weight: 500;\">View theorem statement in API docs →</a>\n </p>\n <div class=\"construction\">\n <strong>Construction:</strong>\n <ul>\n <li>Projected roots: <code>cos(θ + 2πk/N)</code> for k = 0, ..., N-1 <a href=\"./blueprint/sect0002.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §2)</a></li>\n <li>Unscaled polynomial: <code>P(x) = ∏(x - cos(θ + 2πk/N))</code> <a href=\"./blueprint/sect0003.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §3)</a></li>\n <li>Scaled polynomial: <code>S(x) = 2<sup>N-1</sup> · P(x)</code> <a href=\"./blueprint/sect0003.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §3)</a></li>\n <li>Result: <code>S(x) = T<sub>N</sub>(x) + c(θ)</code> <a href=\"./blueprint/sect0001.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §1)</a></li>\n </ul>\n </div>\n </div>\n\n <div class=\"visualization\">\n <figure>\n <video autoplay loop muted playsinline style=\"max-width: 100%; height: auto; border-radius: 8px; box-shadow: 0 4px 12px rgba(0,0,0,0.15);\">\n <source src=\"chebyshev_videos/Chebyshev_N5.mp4\" type=\"video/mp4\">\n Your browser does not support the video tag. Please upgrade to a modern browser.\n </video>\n <figcaption>\n Video animation showing N=5 roots of unity being rotated. The regular pentagon (coral edges) rotates with\n the roots (red points). Vertical projection lines show how roots project to the real axis (pink points),\n which serve as roots for the blue polynomial curve. The displayed polynomial equation updates in real-time,\n showing that the curve's shape remains a 5th-degree Chebyshev polynomial (T₅(x)) while only the constant\n term <em>c</em> changes with rotation angle θ. <strong>Video specifications:</strong> 1080x1920 @ 60fps portrait.\n Rendered with Manim Community Edition.\n </figcaption>\n </figure>\n </div>\n\n <h2>Project Components</h2>\n\n <div class=\"components\">\n <div class=\"component-card\">\n <h3>Formal Verification</h3>\n <p>Complete Lean 4 formalization across 12 modules with full proofs.</p>\n <div class=\"stats\">\n <div class=\"stat\">\n <div class=\"stat-number\">3,548</div>\n <div class=\"stat-label\">Lines of Proof</div>\n </div>\n <div class=\"stat\">\n <div class=\"stat-number\">0</div>\n","line":359,"end_line":408,"proof_status":"formalized"} |
| {"id":"lem:discrete_orthogonality","label":"discrete_orthogonality","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any integer \\(m\\), \\begin{equation} \\sum _{k=0}^{N-1} \\mathrm{e}^{2\\pi i m k/N} = \\begin{cases} N & \\text{if } N \\mid m, \\\\ 0 & \\text{otherwise.} \\end{cases} \\end{equation}","doc_url":"../../docs/find/#doc/sum_primitive_root_pow_mul","latex_link":"sect0007.html#lem:discrete_orthogonality","lean_link":"../../docs/find/#doc/sum_primitive_root_pow_mul","lean_decl_name":"sum_primitive_root_pow_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PowerSums.lean","lean4_code":"lemma sum_primitive_root_pow_mul (N m : ℕ) (hN : 0 < N) (hm : 0 < m) (hm' : m < N) :\n let ω := Complex.exp (2 * π * Complex.I / N)\n ∑ k ∈ Finset.range N, ω ^ (k * m) = 0 := by\n intro ω\n have hN' : N ≠ 0 := Nat.pos_iff_ne_zero.mp hN\n have hω : IsPrimitiveRoot ω N := Complex.isPrimitiveRoot_exp N hN'\n conv_lhs => arg 2; ext k; rw [mul_comm k m, pow_mul]\n have h_ne_one : ω ^ m ≠ 1 := by\n intro h_eq\n have h_div : N ∣ m := (hω.pow_eq_one_iff_dvd m).mp h_eq\n have : N ≤ m := Nat.le_of_dvd hm h_div\n omega\n have h_pow_N : (ω ^ m) ^ N = 1 := by\n calc (ω ^ m) ^ N = ω ^ (m * N) := by rw [← pow_mul]\n _ = ω ^ (N * m) := by rw [mul_comm]\n _ = (ω ^ N) ^ m := by rw [← pow_mul]\n _ = 1 ^ m := by rw [hω.pow_eq_one]\n _ = 1 := one_pow m\n have h_geom : (ω ^ m - 1) * ∑ k ∈ Finset.range N, (ω ^ m) ^ k = (ω ^ m) ^ N - 1 :=\n mul_geom_sum (ω ^ m) N\n rw [h_pow_N] at h_geom\n have h_eq_zero : (ω ^ m - 1) * ∑ k ∈ Finset.range N, (ω ^ m) ^ k = 0 := by\n rw [h_geom]; ring\n exact eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne h_ne_one) h_eq_zero\n\n/-- Sum of cosines at integer multiples of angles vanishes for non-zero integers with |m| < N. -/\n","line":193,"end_line":218,"proof_status":"formalized"} |
| {"id":"lem:sum_cos_m_phi_vanishes","label":"sum_cos_m_phi_vanishes","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For any integers \\(N \\geq 1\\), \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m\\left(\\theta + \\frac{2\\pi k}{N}\\right)\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/sum_cos_multiple_rotated_roots","latex_link":"sect0010.html#lem:sum_cos_m_phi_vanishes","lean_link":"../../docs/find/#doc/sum_cos_multiple_rotated_roots","lean_decl_name":"sum_cos_multiple_rotated_roots","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/TrigonometricIdentities.lean","lean4_code":"lemma sum_cos_multiple_rotated_roots (N : ℕ) (m : ℕ) (θ : ℝ)\n (hN : 0 < N) (hm : 0 < m) (hm' : m < N) :\n ∑ k ∈ Finset.range N, Real.cos (m * (θ + 2 * π * k / N)) = 0 := by\n conv_lhs =>\n arg 2\n ext k\n rw [← Complex.exp_ofReal_mul_I_re (m * (θ + 2 * π * k / N))]\n rw [← Complex.re_sum]\n simp only [Complex.ofReal_mul, Complex.ofReal_add, Complex.ofReal_div, Complex.ofReal_natCast]\n have hN' : N ≠ 0 := Nat.pos_iff_ne_zero.mp hN\n let ζ := Complex.exp (2 * π * I / N)\n have hζ : IsPrimitiveRoot ζ N := Complex.isPrimitiveRoot_exp N hN'\n suffices h_sum_zero : ∑ k ∈ Finset.range N, cexp (↑m * (↑2 * ↑π * ↑k / ↑N) * I) = 0 by\n simp only [mul_add, add_mul]\n conv_lhs =>\n arg 1\n arg 2\n ext x\n rw [Complex.exp_add]\n rw [← Finset.mul_sum]\n have : ∑ i ∈ Finset.range N, cexp (↑m * (↑2 * ↑π * ↑i / ↑N) * I) = 0 := h_sum_zero\n simp [this]\n calc ∑ k ∈ Finset.range N, cexp (↑m * (↑2 * ↑π * ↑k / ↑N) * I)\n = ∑ k ∈ Finset.range N, ζ ^ (m * k) := by\n congr 1 with k\n simp only [ζ]\n rw [← Complex.exp_nat_mul]\n congr 1\n field_simp\n norm_cast\n _ = ∑ k ∈ Finset.range N, (ζ ^ m) ^ k := by\n congr 1 with k\n rw [← pow_mul]\n _ = 0 := by\n have h_ne_one : ζ ^ m ≠ 1 := by\n intro h_eq\n have h_div : N ∣ m := by\n have := hζ.pow_eq_one_iff_dvd m\n exact this.mp h_eq\n have : N ≤ m := Nat.le_of_dvd (by omega) h_div\n omega\n have h_pow_N : (ζ ^ m) ^ N = 1 := by\n rw [← pow_mul, mul_comm m N, pow_mul]\n simp [hζ.pow_eq_one]\n have h_geom : (ζ ^ m - 1) * ∑ k ∈ Finset.range N, (ζ ^ m) ^ k = (ζ ^ m) ^ N - 1 :=\n mul_geom_sum (ζ ^ m) N\n rw [h_pow_N] at h_geom\n have : (ζ ^ m - 1) * ∑ k ∈ Finset.range N, (ζ ^ m) ^ k = 0 := by\n rw [h_geom]; ring\n exact eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne h_ne_one) this\n","line":81,"end_line":130,"proof_status":"formalized"} |
| {"id":"lem:sum_cos_phi_vanishes","label":"sum_cos_phi_vanishes","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For any \\(N \\geq 2\\) and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(\\theta + \\frac{2\\pi k}{N}\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/realProjectionsList_sum","latex_link":"sect0010.html#lem:sum_cos_phi_vanishes","lean_link":"../../docs/find/#doc/realProjectionsList_sum","lean_decl_name":"realProjectionsList_sum","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/RootsOfUnity.lean","lean4_code":"theorem realProjectionsList_sum (N : ℕ) (θ : ℝ) :\n (realProjectionsList N θ).sum = ∑ k ∈ Finset.range N, Real.cos (θ + 2 * π * k / N) := by\n rw [realProjectionsList]\n convert list_range_map_sum_eq_finset_sum N (fun k => Real.cos (θ + 2 * π * ↑k / ↑N)) using 2\n\n/-- The power sum of realProjectionsList equals the Finset power sum. -/\n","line":71,"end_line":76,"proof_status":"formalized"} |
| {"id":"lem:chebyshev_angle_sum_odd","label":"chebyshev_angle_sum_odd","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For odd \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m \\cdot \\frac{(2k + 1)\\pi }{2N}\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/ChebyshevCircles.sum_cos_chebyshev_angles_vanishes","latex_link":"sect0012.html#lem:chebyshev_angle_sum_odd","lean_link":"../../docs/find/#doc/ChebyshevCircles.sum_cos_chebyshev_angles_vanishes","lean_decl_name":"ChebyshevCircles.sum_cos_chebyshev_angles_vanishes","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevOrthogonality.lean","lean4_code":"lemma sum_cos_chebyshev_angles_vanishes (N : ℕ) (m : ℤ) (hN : 0 < N)\n (hm : m ≠ 0) (hm_odd : Odd m) (hm_bound : |m| < 2 * N) :\n ∑ k ∈ range N, Real.cos (m * (2 * k + 1 : ℝ) * π / (2 * N)) = 0 := by\n -- Define the term we're summing\n let f : ℕ → ℝ := fun k ↦ Real.cos (m * (2 * k + 1 : ℝ) * π / (2 * N))\n\n -- We'll show the sum vanishes by pairing terms using symmetry\n -- Key observation: θ_k + θ_{N-1-k} = (2k+1)π/(2N) + (2(N-1-k)+1)π/(2N) = π\n\n -- First, prove the pairing lemma: f(k) + f(N-1-k) = 0\n have pair_sum_zero : ∀ k ∈ range N, f k + f (N - 1 - k) = 0 := by\n intro k hk\n simp only [f]\n rw [mem_range] at hk\n have hk_le : k ≤ N - 1 := Nat.le_pred_of_lt hk\n\n -- First establish the cast identity for (N - 1 - k)\n have cast_eq : ((N - 1 - k) : ℝ) = N - 1 - k := by\n rfl\n\n -- The key insight: (2*(N-1-k)+1)*π/(2N) + (2*k+1)*π/(2N) = π\n -- So cos(m * first_angle) = cos(m*(π - second_angle)) = (-1)^m * cos(m*second_angle)\n\n -- Simplify the angle for N-1-k\n have angle_identity : (2 * (N - 1 - k : ℝ) + 1) * π /\n (2 * N) + (2 * k + 1) * π / (2 * N) = π := by\n have h2N_ne : (2 * N : ℝ) ≠ 0 := by\n have : (0 : ℝ) < N := Nat.cast_pos.mpr hN\n linarith\n rw [← add_div, ← add_mul]\n -- Show the sum of coefficients equals 2N\n have coeff_sum : 2 * (N - 1 - k : ℝ) + 1 + (2 * k + 1) = 2 * N := by\n have : (2 * (N - 1 - k) + 1 : ℕ) + (2 * k + 1) = 2 * N := by omega\n linarith\n rw [coeff_sum]\n field_simp\n\n -- Rewrite second angle as π minus first angle\n have second_angle : (2 * (N - 1 - k : ℝ) + 1) * π / (2 * N) =\n π - (2 * k + 1) * π / (2 * N) := by\n linarith [angle_identity]\n\n -- Substitute and use cosine identity\n -- Need to show the second cosine argument can be rewritten\n have key : m * (2 * ((N - 1 - k) : ℝ) + 1) * π / (2 * N) =\n m * (π - (2 * k + 1) * π / (2 * N)) := by\n calc m * (2 * (N - 1 - k : ℝ) + 1) * π / (2 * N)\n = m * ((2 * (N - 1 - k : ℝ) + 1) * π / (2 * N)) := by ring\n _ = m * (π - (2 * k + 1) * π / (2 * N)) := by rw [second_angle]\n\n","line":57,"end_line":106,"proof_status":"formalized"} |
| {"id":"lem:chebyshev_angle_sum_even","label":"chebyshev_angle_sum_even","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For even \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m \\cdot \\frac{(2k + 1)\\pi }{2N}\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/ChebyshevCircles.binomial_terms_vanish_chebyshev","latex_link":"sect0012.html#lem:chebyshev_angle_sum_even","lean_link":"../../docs/find/#doc/ChebyshevCircles.binomial_terms_vanish_chebyshev","lean_decl_name":"ChebyshevCircles.binomial_terms_vanish_chebyshev","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevOrthogonality.lean","lean4_code":"lemma binomial_terms_vanish_chebyshev (N j r : ℕ) (hN : 0 < N) (hj' : j < N)\n (hr : r ∈ range (j + 1)) (hr_neq : (2 * r - j : ℤ) ≠ 0) :\n ∑ k ∈ range N, Real.cos ((2 * r - j : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N)) = 0 := by\n -- We need to show:\n -- 1. (2r - j) is odd\n -- 2. |2r - j| < 2N\n -- Then apply sum_cos_chebyshev_angles_vanishes\n\n -- First establish the bound: r ≤ j\n have hr_le : r ≤ j := by\n simp only [mem_range] at hr\n omega\n\n -- Split by parity of j\n by_cases h_parity : Odd j\n · -- Case 1: j is odd, so (2r - j) is odd\n have h_odd : Odd (2 * r - j : ℤ) := by\n obtain ⟨m, hm⟩ := h_parity\n use (r : ℤ) - m - 1\n have : (j : ℤ) = 2 * m + 1 := by exact_mod_cast hm\n calc (2 * r - j : ℤ)\n _ = 2 * (r : ℤ) - (2 * m + 1) := by rw [this]\n _ = 2 * ((r : ℤ) - m - 1) + 1 := by ring\n\n -- Prove |2r - j| < 2N\n have h_bound : |(2 * r - j : ℤ)| < 2 * N := by\n by_cases hr_case : 2 * r ≥ j\n · rw [abs_of_nonneg (Int.sub_nonneg_of_le (by omega : (j : ℤ) ≤ (2 * r : ℤ)))]\n calc (2 * r - j : ℤ)\n _ ≤ (2 * j - j : ℤ) := by omega\n _ = (j : ℤ) := by ring\n _ < (N : ℤ) := by omega\n _ < (2 * N : ℤ) := by omega\n · push_neg at hr_case\n rw [abs_of_neg (Int.sub_neg_of_lt (by omega : (2 * r : ℤ) < (j : ℤ)))]\n calc -(2 * r - j : ℤ)\n _ = (j - 2 * r : ℤ) := by ring\n _ ≤ (j : ℤ) := by omega\n _ < (N : ℤ) := by omega\n _ < (2 * N : ℤ) := by omega\n\n -- Apply odd vanishing lemma\n convert sum_cos_chebyshev_angles_vanishes N (2 * r - j) hN hr_neq h_odd h_bound using 2\n\n · -- Case 2: j is even, so (2r - j) is even\n have j_even : Even j := Nat.not_odd_iff_even.mp h_parity\n have h_even : Even (2 * r - j : ℤ) := by\n obtain ⟨m, hm⟩ := j_even\n use (r : ℤ) - m\n have : (j : ℤ) = (m + m : ℕ) := by simp [hm]\n","line":410,"end_line":459,"proof_status":"formalized"} |
| {"id":"thm:newton_identities","label":"newton_identities","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"The power sums and elementary symmetric polynomials are related by: \\begin{equation} k \\cdot e_k = \\sum _{i=1}^k (-1)^{i-1} e_{k-i} p_i, \\quad k = 1, 2, \\ldots , N, \\end{equation}","doc_url":"../../docs/find/#doc/esymm_eq_of_psum_eq","latex_link":"sect0008.html#thm:newton_identities","lean_link":"../../docs/find/#doc/esymm_eq_of_psum_eq","lean_decl_name":"esymm_eq_of_psum_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/NewtonIdentities.lean","lean4_code":"lemma esymm_eq_of_psum_eq (s t : Multiset ℝ) (h_card : s.card = t.card)\n (h_psum : ∀ j, 0 < j → j < s.card → (s.map (· ^ j)).sum = (t.map (· ^ j)).sum) :\n ∀ m, m < s.card → s.esymm m = t.esymm m := by\n intro m hm\n -- We prove by strong induction on m\n induction m using Nat.strong_induction_on with\n | h m IH =>\n by_cases hm_zero : m = 0\n · -- Base case: esymm 0 = 1 for any multiset\n rw [hm_zero]\n simp [Multiset.esymm]\n · -- Inductive case: m > 0, use Newton's identity\n have hm_pos : 0 < m := Nat.pos_of_ne_zero hm_zero\n -- Apply Newton's identity to both s and t\n have h_newton_s := multiset_newton_identity s m hm_pos\n have h_newton_t := multiset_newton_identity t m hm_pos\n -- The RHS of Newton's identity should be equal for s and t\n have h_rhs_eq : (Finset.antidiagonal m).sum (fun a =>\n if a.1 < m then (-1)^a.1 * s.esymm a.1 * (s.map (fun x => x ^ a.2)).sum\n else 0) =\n (Finset.antidiagonal m).sum (fun a =>\n if a.1 < m then (-1)^a.1 * t.esymm a.1 * (t.map (fun x => x ^ a.2)).sum\n else 0) := by\n apply Finset.sum_congr rfl\n intro a ha\n -- a ∈ antidiagonal m means a.1 + a.2 = m\n have h_sum : a.1 + a.2 = m := Finset.mem_antidiagonal.mp ha\n split_ifs with ha_lt\n · -- When a.1 < m, show the terms are equal\n congr 1\n · congr 1 -- esymm a.1 equal by IH\n have ha1_bound : a.1 < s.card := by omega\n exact IH a.1 ha_lt ha1_bound\n · -- Power sums equal by hypothesis\n by_cases ha2_zero : a.2 = 0\n · -- When a.2 = 0, both power sums are cardinalities\n simp only [ha2_zero, pow_zero]\n -- Simplify x^0 = 1, then sum of 1s equals card\n convert_to (s.card : ℝ) = (t.card : ℝ) using 1 <;>\n simp\n exact_mod_cast h_card\n · have ha2_pos : 0 < a.2 := Nat.pos_of_ne_zero ha2_zero\n have ha2_bound : a.2 < s.card := by omega\n exact h_psum a.2 ha2_pos ha2_bound\n · rfl\n -- Now use Newton's identity to extract esymm m\n have h_eq : (m : ℝ) * s.esymm m = (m : ℝ) * t.esymm m := by\n rw [h_newton_s, h_newton_t, h_rhs_eq]\n -- Cancel the factor of m\n have hm_ne_zero : (m : ℝ) ≠ 0 := by\n","line":132,"end_line":181,"proof_status":"formalized"} |
| {"id":"lem:power_sum_invariance","label":"power_sum_invariance","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:sum_cos_m_phi_vanishes"],"nlp_text":"For any integers \\(N \\geq 1\\), \\(j\\) with \\(1 \\leq j {\\lt} N\\), and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos ^j\\left(\\theta + \\frac{2\\pi k}{N}\\right) \\end{equation}","doc_url":"../../docs/find/#doc/cos_cube_formula","latex_link":"sect0020.html#lem:power_sum_invariance","lean_link":"../../docs/find/#doc/cos_cube_formula","lean_decl_name":"cos_cube_formula","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/TrigonometricIdentities.lean","lean4_code":"lemma cos_cube_formula (x : ℝ) :\n Real.cos x ^ 3 = (Real.cos (3 * x) + 3 * Real.cos x) / 4 := by\n linarith [Real.cos_three_mul x]\n\n","line":133,"end_line":136,"proof_status":"formalized"} |
| {"id":"lem:chebyshev_roots","label":"chebyshev_roots","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:chebyshev_angle_sum_odd","lem:chebyshev_angle_sum_even","lem:chebyshev_roots_characterization"],"nlp_text":"Let \\(\\xi _k = \\cos ((2k + 1)\\pi /(2N))\\) for \\(k = 0, \\ldots , N-1\\) denote the roots of \\(T_N(x)\\). Then for any \\(1 \\leq j {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\xi _k^j = \\begin{cases} \\frac{N}{2^j} \\binom {j}{j/2} & \\text{if } j \\text{ is even}, \\\\ 0 & \\text{if } j \\text{ is odd.} \\end{cases} \\end{equation}","doc_url":"../../docs/find/#doc/ChebyshevCircles.sum_cos_pow_chebyshev_binomial","latex_link":"sect0027.html#lem:chebyshev_roots","lean_link":"../../docs/find/#doc/ChebyshevCircles.sum_cos_pow_chebyshev_binomial","lean_decl_name":"ChebyshevCircles.sum_cos_pow_chebyshev_binomial","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevOrthogonality.lean","lean4_code":"lemma sum_cos_pow_chebyshev_binomial (N j : ℕ) (_hN : 0 < N) (_hj : 0 < j) (_hj' : j < 2 * N) :\n ∑ k ∈ range N, (Real.cos ((2 * k + 1 : ℝ) * π / (2 * N))) ^ j =\n (2 : ℝ) ^ (-(j : ℤ)) * ∑ r ∈ range (j + 1), (j.choose r : ℝ) *\n ∑ k ∈ range N, Real.cos ((2 * r - j : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N)) := by\n -- Follows sum_cos_pow_eq_sum_binomial from PowerSums.lean\n simp only [cos_as_exp_chebyshev]\n\n -- Pull power inside re: (z.re)^j = (z^j).re when z is real\n have h_re_pow : ∀ k, ((Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))) / 2).re ^ j =\n (((Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))) / 2) ^ j).re := by\n intro k\n set z := (Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))) / 2 with z_def\n have h_real : z.im = 0 := by\n rw [z_def]\n have h1 : (Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))).im = 0 := by\n rw [Complex.add_im, Complex.exp_ofReal_mul_I_im]\n have : (Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))).im =\n Real.sin (-((2 * k + 1 : ℝ) * π / (2 * N))) := by\n rw [show -(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) =\n (↑(-((2 * k + 1 : ℝ) * π / (2 * N))) : ℂ) * Complex.I by\n apply Complex.ext <;> simp]\n exact Complex.exp_ofReal_mul_I_im _\n rw [this, Real.sin_neg]\n ring\n rw [Complex.div_im, h1]\n simp\n have h_eq : z = (z.re : ℂ) := Complex.ext rfl h_real\n calc z.re ^ j\n _ = (z.re : ℂ).re ^ j := by simp\n _ = ((z.re : ℂ) ^ j).re := by simp [← Complex.ofReal_pow]\n _ = (z ^ j).re := by rw [← h_eq]\n\n conv_lhs =>\n arg 2\n ext k\n rw [h_re_pow k]\n rw [← Complex.re_sum]\n\n simp only [div_pow]\n\n rw [show ∑ x ∈ range N, (Complex.exp (↑((2 * ↑x + 1) * π / (2 * ↑N)) * Complex.I) +\n Complex.exp (-(↑((2 * ↑x + 1) * π / (2 * ↑N)) * Complex.I))) ^ j / (2 : ℂ) ^ j =\n ∑ x ∈ range N, (∑ r ∈ range (j + 1), (j.choose r : ℂ) *\n Complex.exp ((2 * r - j : ℤ) * ↑((2 * ↑x + 1) * π / (2 * ↑N)) * Complex.I)) / (2 : ℂ) ^ j by\n apply Finset.sum_congr rfl\n intro i _\n","line":194,"end_line":243,"proof_status":"formalized"} |
| {"id":"lem:esymm_invariant","label":"esymm_invariant","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["thm:newton_identities","lem:power_sum_invariance"],"nlp_text":"Let \\(r_k(\\theta ) = \\cos (\\theta + 2\\pi k/N)\\) for \\(k = 0, \\ldots , N-1\\). Then for any \\(m\\) with \\(1 \\leq m \\leq N\\), the elementary symmetric polynomial \\begin{equation} e_m(r_0(\\theta ), \\ldots , r_{N-1}(\\theta )) \\end{equation}","doc_url":"../../docs/find/#doc/esymm_rotated_roots_invariant","latex_link":"sect0022.html#lem:esymm_invariant","lean_link":"../../docs/find/#doc/esymm_rotated_roots_invariant","lean_decl_name":"esymm_rotated_roots_invariant","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/NewtonIdentities.lean","lean4_code":"lemma esymm_rotated_roots_invariant (N : ℕ) (m : ℕ) (θ₁ θ₂ : ℝ)\n (hN : 0 < N) (hm : 0 < m) (hm' : m < N) :\n let l1 := (realProjectionsList N θ₁ : Multiset ℝ)\n let l2 := (realProjectionsList N θ₂ : Multiset ℝ)\n l1.esymm m = l2.esymm m := by\n intro l1 l2\n induction m using Nat.strong_induction_on with\n | h k IH =>\n cases k with\n | zero => omega\n | succ k' =>\n cases k' with\n | zero =>\n -- N ≥ 2, so sum of cosines at N equally spaced angles = 0\n -- esymm 1 is the sum, and both sums equal 0\n rw [multiset_esymm_one_eq_sum, multiset_esymm_one_eq_sum]\n simp only [l1, l2]\n have hN2 : 2 ≤ N := by omega\n rw [multiset_coe_realProjectionsList_sum, multiset_coe_realProjectionsList_sum]\n rw [sum_cos_roots_of_unity N θ₁ hN2, sum_cos_roots_of_unity N θ₂ hN2]\n | succ k'' =>\n -- Use Newton's identity to express esymm (k''+2) in terms of smaller esymm and power sums\n have h_newton_l1 := multiset_newton_identity l1 (k'' + 2) (by omega)\n have h_newton_l2 := multiset_newton_identity l2 (k'' + 2) (by omega)\n\n -- The RHS of Newton's identity is the same for l1 and l2\n have h_rhs_eq : (Finset.antidiagonal (k'' + 2)).sum (fun a =>\n if a.1 < k'' + 2 then (-1)^a.1 * l1.esymm a.1 * (l1.map (fun x => x ^ a.2)).sum\n else 0) =\n (Finset.antidiagonal (k'' + 2)).sum (fun a =>\n if a.1 < k'' + 2 then (-1)^a.1 * l2.esymm a.1 * (l2.map (fun x => x ^ a.2)).sum\n else 0) := by\n -- Each term in the sum is equal\n apply Finset.sum_congr rfl\n intro a ha_mem\n -- a ∈ antidiagonal (k'' + 2), so a.1 + a.2 = k'' + 2\n have h_sum : a.1 + a.2 = k'' + 2 := Finset.mem_antidiagonal.mp ha_mem\n split_ifs with ha\n · -- When a.1 < k'' + 2, we need to show the terms are equal\n congr 1\n · -- esymm a.1 is invariant by IH\n by_cases ha0 : a.1 = 0\n · simp only [ha0, pow_zero, one_mul]\n -- esymm 0 = 1 for any multiset\n simp only [Multiset.esymm, Multiset.powersetCard_zero_left]\n · have ha_pos : 0 < a.1 := Nat.pos_of_ne_zero ha0\n -- Need to show a.1 < N\n have ha_N : a.1 < N := by\n calc a.1\n _ < k'' + 2 := ha\n","line":238,"end_line":287,"proof_status":"formalized"} |
| {"id":"cor:power_sums_determine_coefficients","label":"power_sums_determine_coefficients","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["thm:newton_identities"],"nlp_text":"Let \\(\\alpha _1, \\ldots , \\alpha _N\\) and \\(\\beta _1, \\ldots , \\beta _N\\) be two collections of numbers. If \\begin{equation} \\sum _{i=1}^N \\alpha _i^j = \\sum _{i=1}^N \\beta _i^j \\quad \\text{for all } j = 1, 2, \\ldots , N, \\end{equation}","doc_url":"../../docs/find/#doc/Multiset.prod_X_sub_C_coeff","latex_link":"sect0008.html#cor:power_sums_determine_coefficients","lean_link":"../../docs/find/#doc/Multiset.prod_X_sub_C_coeff","lean_decl_name":"Multiset.prod_X_sub_C_coeff","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":null,"lean4_code":null,"line":30,"end_line":30,"proof_status":"formalized"} |
| {"id":"thm:power_sum_equality","label":"power_sum_equality","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:power_sum_invariance","lem:chebyshev_roots"],"nlp_text":"For \\(1 \\leq j {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos ^j\\left(\\theta + \\frac{2\\pi k}{N}\\right) = \\sum _{k=0}^{N-1} \\cos ^j\\left(\\frac{(2k + 1)\\pi }{2N}\\right). \\end{equation}","doc_url":"../../docs/find/#doc/general_powersum_equality","latex_link":"sect0027.html#thm:power_sum_equality","lean_link":"../../docs/find/#doc/general_powersum_equality","lean_decl_name":"general_powersum_equality","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PowerSumEquality.lean","lean4_code":"theorem general_powersum_equality (N j : ℕ) (hN : 0 < N) (hj : 0 < j) (hj' : j < N) :\n ∑ k ∈ Finset.range N, (Real.cos (2 * π * k / N)) ^ j =\n ∑ k ∈ Finset.range N, (Real.cos ((2 * k + 1 : ℝ) * π / (2 * N))) ^ j := by\n -- Both sums equal the same closed-form expression\n rw [rotated_roots_powersum_value N j hN hj hj']\n rw [chebyshev_roots_powersum_value N j hN hj hj']\n\nend PowerSumEquality\n\n","line":264,"end_line":272,"proof_status":"formalized"} |
| {"id":"thm:coefficients_independent","label":"coefficients_independent","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:esymm_invariant","cor:power_sums_determine_coefficients"],"nlp_text":"For any \\(N \\geq 1\\) and any \\(k\\) with \\(1 \\leq k \\leq N\\), the coefficient of \\(x^k\\) in \\(S_N(x; \\theta )\\) is independent of \\(\\theta \\). (See Theorem 5.1 in ../paper/chebyshev_circles.pdfthe paper)","doc_url":"../../docs/find/#doc/constant_term_only_varies","latex_link":"sect0022.html#thm:coefficients_independent","lean_link":"../../docs/find/#doc/constant_term_only_varies","lean_decl_name":"constant_term_only_varies","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PolynomialProperties.lean","lean4_code":"theorem constant_term_only_varies (N : ℕ) (θ₁ θ₂ : ℝ) (k : ℕ) (hN : 0 < N)\n (hk : 0 < k) : (scaledPolynomial N θ₁).coeff k = (scaledPolynomial N θ₂).coeff k := by\n unfold scaledPolynomial unscaledPolynomial polynomialFromRealRoots\n simp only [coeff_C_mul]\n congr 1\n -- Convert to multiset form\n rw [list_foldr_eq_multiset_prod, list_foldr_eq_multiset_prod]\n -- Factor out the card lemma\n have h_card (θ : ℝ) : ((realProjectionsList N θ : Multiset ℝ)).card = N := by\n rw [Multiset.coe_card, card_realProjectionsList]\n -- Split cases\n by_cases hk_le : k ≤ N\n · -- k ≤ N: use Vieta's formula\n rw [Multiset.prod_X_sub_C_coeff (realProjectionsList N θ₁ : Multiset ℝ) (by rwa [h_card]),\n Multiset.prod_X_sub_C_coeff (realProjectionsList N θ₂ : Multiset ℝ) (by rwa [h_card])]\n congr 1\n · rw [h_card, h_card]\n · by_cases hk_eq : k = N\n · rw [hk_eq, h_card, h_card]; norm_num [Multiset.esymm, Multiset.powersetCard_zero_left]\n · rw [h_card, h_card]\n exact esymm_rotated_roots_invariant N (N - k) θ₁ θ₂ hN (by omega) (by omega)\n · -- k > N: both coefficients are 0\n have deg (θ : ℝ) : (Multiset.map (fun r => X - C r)\n (realProjectionsList N θ : Multiset ℝ)).prod.natDegree = N := by\n rw [Polynomial.natDegree_multiset_prod_X_sub_C_eq_card, h_card]\n rw [Polynomial.coeff_eq_zero_of_natDegree_lt, Polynomial.coeff_eq_zero_of_natDegree_lt]\n · rw [deg]; omega\n · rw [deg]; omega\n\n/-- The Chebyshev polynomial T_N has degree N for N ≥ 1. -/\n","line":39,"end_line":68,"proof_status":"formalized"} |
| {"id":"prop:chebyshev_properties","label":"chebyshev_properties","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The Chebyshev polynomials satisfy the following properties: Recurrence: \\(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\\) for \\(n \\geq 1\\). Leading coefficient: The leading coefficient of \\(T_N(x)\\) is \\(2^{N-1}\\) for \\(N \\geq 1\\). Roots: The roots of \\(T_N(x)\\) are \"> \\begin{equation} \\label{eq:chebyshev_roots} x_k = \\cos \\left(\\frac{(2k + 1)\\pi }{2N}\\right), \\quad k = 0, 1, \\ldots , N-1. \\end{equation}","doc_url":"../../docs/find/#doc/Polynomial.Chebyshev.T","latex_link":"sect0006.html#prop:chebyshev_properties","lean_link":"../../docs/find/#doc/Polynomial.Chebyshev.T","lean_decl_name":"Polynomial.Chebyshev.T","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":null,"lean4_code":null,"line":31,"end_line":31,"proof_status":"formalized"} |
| {"id":"lem:chebyshev_roots_characterization","label":"chebyshev_roots_characterization","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["prop:chebyshev_properties"],"nlp_text":"The roots of the Chebyshev polynomial \\(T_N(x)\\) are \\begin{equation} \\xi _k = \\cos \\left(\\frac{(2k + 1)\\pi }{2N}\\right), \\quad k = 0, 1, \\ldots , N-1. \\end{equation}","doc_url":"../../docs/find/#doc/chebyshevRoot_in_Icc","latex_link":"sect0025.html#lem:chebyshev_roots_characterization","lean_link":"../../docs/find/#doc/chebyshevRoot_in_Icc","lean_decl_name":"chebyshevRoot_in_Icc","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevRoots.lean","lean4_code":"lemma chebyshevRoot_in_Icc (N k : ℕ) : chebyshevRoot N k ∈ Set.Icc (-1) 1 := by\n unfold chebyshevRoot\n exact Real.cos_mem_Icc _\n\n/-- Backward direction: T_N(cos((2k+1)π/(2N))) = 0 -/\n","line":37,"end_line":41,"proof_status":"formalized"} |
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