openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /QuasinilpotentCommuting.lean
| import OAI.Analysis.IrrationalRotation.Model.Analytic | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Continuity | |
| import Mathlib.Data.Nat.Choose.Sum | |
| import Mathlib.Tactic | |
| namespace OAI | |
| noncomputable section | |
| open Filter | |
| open scoped Topology BigOperators | |
| namespace Invariant233 | |
| theorem exists_geometric_bound_of_root_tendsto_zero (u : ℕ → ℝ) | |
| (hu : ∀ n, 0 ≤ u n) | |
| (hroot : Tendsto (fun n : ℕ => (u n) ^ (1 / (n : ℝ))) atTop (𝓝 0)) | |
| (r : ℝ) (hr : 0 < r) : | |
| ∃ C : ℝ, 0 < C ∧ ∀ n, u n ≤ C * r ^ n := by | |
| have hev : ∀ᶠ n : ℕ in atTop, (u n) ^ (1 / (n : ℝ)) < r := | |
| (tendsto_order.mp hroot).2 r hr | |
| obtain ⟨N, hN⟩ := eventually_atTop.1 (hev.and (eventually_gt_atTop 0)) | |
| let C : ℝ := 1 + ∑ k ∈ Finset.range N, u k / r ^ k | |
| have hsum : 0 ≤ ∑ k ∈ Finset.range N, u k / r ^ k := | |
| Finset.sum_nonneg fun k _ => div_nonneg (hu k) (pow_nonneg hr.le k) | |
| have hC : 0 < C := by dsimp [C]; linarith | |
| refine ⟨C, hC, fun n => ?_⟩ | |
| by_cases hn : n < N | |
| · have hle : u n / r ^ n ≤ ∑ k ∈ Finset.range N, u k / r ^ k := | |
| Finset.single_le_sum (fun k _ => div_nonneg (hu k) (pow_nonneg hr.le k)) | |
| (Finset.mem_range.mpr hn) | |
| have hleC : u n / r ^ n ≤ C := by dsimp [C]; linarith | |
| exact (div_le_iff₀ (pow_pos hr n)).mp hleC | |
| · obtain ⟨hsmall, hnpos⟩ := hN n (Nat.le_of_not_gt hn) | |
| have hpow := pow_le_pow_left₀ (Real.rpow_nonneg (hu n) _) hsmall.le n | |
| rw [one_div, Real.rpow_inv_natCast_pow (hu n) (Nat.ne_of_gt hnpos)] at hpow | |
| exact hpow.trans (le_mul_of_one_le_left (pow_nonneg hr.le n) (by dsimp [C]; linarith)) | |
| theorem root_tendsto_zero_of_geometric_bounds (u : ℕ → ℝ) | |
| (hu : ∀ n, 0 ≤ u n) | |
| (hbound : ∀ r : ℝ, 0 < r → ∃ C : ℝ, 0 < C ∧ ∀ n, u n ≤ C * r ^ n) : | |
| Tendsto (fun n : ℕ => (u n) ^ (1 / (n : ℝ))) atTop (𝓝 0) := by | |
| apply tendsto_order.mpr | |
| constructor | |
| · intro a ha | |
| exact Eventually.of_forall fun n => ha.trans_le (Real.rpow_nonneg (hu n) _) | |
| · intro ε hε | |
| obtain ⟨C, hC, hboundC⟩ := hbound (ε / 2) (by positivity) | |
| have hCroot : Tendsto (fun n : ℕ => C ^ (1 / (n : ℝ))) atTop (𝓝 1) := by | |
| simpa [Function.comp_def] using (Real.continuousAt_const_rpow hC.ne').tendsto.comp | |
| (tendsto_one_div_atTop_nhds_zero_nat : | |
| Tendsto (fun n : ℕ => 1 / (n : ℝ)) atTop (𝓝 0)) | |
| have hev : ∀ᶠ n : ℕ in atTop, C ^ (1 / (n : ℝ)) < 2 := | |
| (tendsto_order.mp hCroot).2 2 (by norm_num) | |
| filter_upwards [hev, eventually_gt_atTop (0 : ℕ)] with n hn hnpos | |
| have hle := Real.rpow_le_rpow (hu n) (hboundC n) | |
| (by positivity : 0 ≤ 1 / (n : ℝ)) | |
| rw [Real.mul_rpow hC.le (pow_nonneg (by positivity : 0 ≤ ε / 2) n), | |
| one_div, Real.pow_rpow_inv_natCast (by positivity : 0 ≤ ε / 2) | |
| (Nat.ne_of_gt hnpos)] at hle | |
| have hlt := mul_lt_mul_of_pos_right hn (by positivity : 0 < ε / 2) | |
| have hlt' : C ^ ((n : ℝ)⁻¹) * (ε / 2) < ε := by | |
| simpa only [one_div, mul_div_cancel₀ _ (by norm_num : (2 : ℝ) ≠ 0)] using hlt | |
| simpa only [one_div] using hle.trans_lt hlt' | |
| variable {A : Type*} [NormedRing A] | |
| theorem NormQuasinilpotent.geometric_bound {a : A} (ha : NormQuasinilpotent a) | |
| (r : ℝ) (hr : 0 < r) : ∃ C : ℝ, 0 < C ∧ ∀ n, ‖a ^ n‖ ≤ C * r ^ n := | |
| exists_geometric_bound_of_root_tendsto_zero (fun n => ‖a ^ n‖) | |
| (fun _ => norm_nonneg _) ha r hr | |
| theorem NormQuasinilpotent.mul_of_commute {a b : A} | |
| (ha : NormQuasinilpotent a) (hb : NormQuasinilpotent b) (hab : Commute a b) : | |
| NormQuasinilpotent (a * b) := by | |
| apply root_tendsto_zero_of_geometric_bounds _ (fun _ => norm_nonneg _) | |
| intro r hr | |
| obtain ⟨Ca, hCa, haC⟩ := ha.geometric_bound r hr | |
| obtain ⟨Cb, hCb, hbC⟩ := hb.geometric_bound 1 zero_lt_one | |
| refine ⟨Ca * Cb, mul_pos hCa hCb, fun n => ?_⟩ | |
| rw [hab.mul_pow] | |
| calc | |
| ‖a ^ n * b ^ n‖ ≤ ‖a ^ n‖ * ‖b ^ n‖ := norm_mul_le _ _ | |
| _ ≤ (Ca * r ^ n) * (Cb * 1 ^ n) := | |
| mul_le_mul (haC n) (hbC n) (norm_nonneg _) (by positivity) | |
| _ = Ca * Cb * r ^ n := by ring | |
| theorem norm_add_pow_le_of_geometric_bounds {a b : A} (hab : Commute a b) | |
| {Ca Cb r : ℝ} (hCa : 0 ≤ Ca) (_hCb : 0 ≤ Cb) (hr : 0 ≤ r) | |
| (ha : ∀ n, ‖a ^ n‖ ≤ Ca * r ^ n) | |
| (hb : ∀ n, ‖b ^ n‖ ≤ Cb * r ^ n) (n : ℕ) : | |
| ‖(a + b) ^ n‖ ≤ Ca * Cb * (2 * r) ^ n := by | |
| have hexp : (a + b) ^ n = ∑ k ∈ Finset.range (n + 1), | |
| n.choose k • (a ^ k * b ^ (n - k)) := by | |
| simpa only [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] using hab.add_pow' n | |
| rw [hexp] | |
| calc | |
| ‖∑ k ∈ Finset.range (n + 1), n.choose k • (a ^ k * b ^ (n - k))‖ | |
| ≤ ∑ k ∈ Finset.range (n + 1), ‖n.choose k • (a ^ k * b ^ (n - k))‖ := | |
| norm_sum_le _ _ | |
| _ ≤ ∑ k ∈ Finset.range (n + 1), | |
| (n.choose k : ℝ) * ((Ca * r ^ k) * (Cb * r ^ (n - k))) := by | |
| apply Finset.sum_le_sum | |
| intro k hk | |
| exact norm_nsmul_le.trans (mul_le_mul_of_nonneg_left | |
| ((norm_mul_le _ _).trans (mul_le_mul (ha k) (hb (n - k)) | |
| (norm_nonneg _) (mul_nonneg hCa (pow_nonneg hr _)))) (Nat.cast_nonneg _)) | |
| _ = Ca * Cb * (∑ k ∈ Finset.range (n + 1), | |
| r ^ k * r ^ (n - k) * (n.choose k : ℝ)) := by | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro k hk | |
| ring | |
| _ = Ca * Cb * (2 * r) ^ n := by rw [← add_pow]; congr 2; ring | |
| theorem NormQuasinilpotent.add_of_commute {a b : A} | |
| (ha : NormQuasinilpotent a) (hb : NormQuasinilpotent b) (hab : Commute a b) : | |
| NormQuasinilpotent (a + b) := by | |
| apply root_tendsto_zero_of_geometric_bounds _ (fun _ => norm_nonneg _) | |
| intro r hr | |
| obtain ⟨Ca, hCa, haC⟩ := ha.geometric_bound (r / 2) (by positivity) | |
| obtain ⟨Cb, hCb, hbC⟩ := hb.geometric_bound (r / 2) (by positivity) | |
| refine ⟨Ca * Cb, mul_pos hCa hCb, fun n => ?_⟩ | |
| convert norm_add_pow_le_of_geometric_bounds hab hCa.le hCb.le | |
| (by positivity : 0 ≤ r / 2) haC hbC n using 1 | |
| ring | |
| end Invariant233 | |
| end | |
| end OAI | |