openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /PhaseFailureSummability2.lean
| import Mathlib | |
| namespace OAI | |
| namespace Invariant233.PhaseFailureAudit | |
| open Filter MeasureTheory | |
| open scoped ENNReal | |
| theorem phase_failure_bound_eq {m : ℝ} (hm : 0 < m) : | |
| 2 * m ^ ((4 / 5 : ℝ) * m) * (2 : ℝ) ^ m * | |
| (10 * m ^ (-(19 / 20 : ℝ))) ^ ((19 / 20 : ℝ) * m) = | |
| 2 * Real.exp (-(41 / 400 : ℝ) * m * Real.log m + | |
| (Real.log 2 + (19 / 20 : ℝ) * Real.log 10) * m) := by | |
| rw [Real.rpow_def_of_pos hm, Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)] | |
| rw [Real.rpow_def_of_pos (by positivity : 0 < 10 * m ^ (-(19 / 20 : ℝ)))] | |
| rw [Real.log_mul (by norm_num : (10 : ℝ) ≠ 0) | |
| (ne_of_gt (Real.rpow_pos_of_pos hm _)), Real.log_rpow hm] | |
| calc | |
| _ = 2 * (Real.exp (Real.log m * ((4 / 5 : ℝ) * m)) * | |
| Real.exp (Real.log 2 * m) * | |
| Real.exp ((Real.log 10 + (-(19 / 20 : ℝ)) * Real.log m) * | |
| ((19 / 20 : ℝ) * m))) := by ring | |
| _ = _ := by | |
| rw [← Real.exp_add, ← Real.exp_add] | |
| congr 2 | |
| ring | |
| theorem summable_exp_neg_mul_log {c C : ℝ} (hc : 0 < c) | |
| (m : ℕ → ℝ) (hm : ∀ j : ℕ, (j : ℝ) ≤ m j) : | |
| Summable (fun j => Real.exp (-c * m j * Real.log (m j) + C * m j)) := by | |
| have hmtop : Tendsto m atTop atTop := | |
| tendsto_atTop_mono hm tendsto_natCast_atTop_atTop | |
| have hlogtop : Tendsto (fun j => Real.log (m j)) atTop atTop := | |
| Real.tendsto_log_atTop.comp hmtop | |
| have hlog : ∀ᶠ j in atTop, (C + 1) / c ≤ Real.log (m j) := | |
| hlogtop.eventually_ge_atTop _ | |
| have hs : Summable (fun j => Real.exp (-m j)) := by | |
| simpa only [neg_one_mul] using | |
| (Real.summable_exp_nat_mul_of_ge (by norm_num : (-1 : ℝ) < 0) hm) | |
| apply hs.of_norm_bounded_eventually_nat | |
| filter_upwards [hlog] with j hj | |
| rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] | |
| apply Real.exp_le_exp.mpr | |
| have hm0 : 0 ≤ m j := le_trans (Nat.cast_nonneg j) (hm j) | |
| have hcj : C + 1 ≤ c * Real.log (m j) := by | |
| have := (div_le_iff₀ hc).mp hj | |
| nlinarith | |
| have hmul := mul_le_mul_of_nonneg_right hcj hm0 | |
| nlinarith | |
| theorem summable_phase_failure_bound (m : ℕ → ℝ) | |
| (hmpos : ∀ j, 0 < m j) (hm : ∀ j : ℕ, (j : ℝ) ≤ m j) : | |
| Summable (fun j => 2 * (m j) ^ ((4 / 5 : ℝ) * m j) * (2 : ℝ) ^ m j * | |
| (10 * (m j) ^ (-(19 / 20 : ℝ))) ^ ((19 / 20 : ℝ) * m j)) := by | |
| simp_rw [phase_failure_bound_eq (hmpos _)] | |
| exact (summable_exp_neg_mul_log (by norm_num : (0 : ℝ) < 41 / 400) | |
| m hm).mul_left 2 | |
| theorem summable_phase_failure_bound_nat (m : ℕ → ℕ) | |
| (hm : ∀ j, 2 ^ j ≤ m j) : | |
| Summable (fun j => 2 * (m j : ℝ) ^ ((4 / 5 : ℝ) * m j) * (2 : ℝ) ^ m j * | |
| (10 * (m j : ℝ) ^ (-(19 / 20 : ℝ))) ^ ((19 / 20 : ℝ) * m j)) := by | |
| have hmpos : ∀ j, 0 < (m j : ℝ) := by | |
| intro j | |
| exact_mod_cast lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hm j) | |
| have hmindex : ∀ j : ℕ, (j : ℝ) ≤ (m j : ℝ) := by | |
| intro j | |
| exact_mod_cast (le_trans (Nat.le_of_lt Nat.lt_two_pow_self) (hm j)) | |
| simpa only [Real.rpow_natCast] using | |
| summable_phase_failure_bound (fun j => (m j : ℝ)) hmpos hmindex | |
| theorem ae_eventually_notMem_of_phase_failure_bound | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (bad : ℕ → Set Ω) | |
| (m : ℕ → ℕ) (hm : ∀ j, 2 ^ j ≤ m j) | |
| (hbad : ∀ j, μ (bad j) ≤ ENNReal.ofReal | |
| (2 * (m j : ℝ) ^ ((4 / 5 : ℝ) * m j) * (2 : ℝ) ^ m j * | |
| (10 * (m j : ℝ) ^ (-(19 / 20 : ℝ))) ^ ((19 / 20 : ℝ) * m j))) : | |
| ∀ᵐ ω ∂μ, ∀ᶠ j in atTop, ω ∉ bad j := by | |
| apply MeasureTheory.ae_eventually_notMem | |
| exact ne_top_of_le_ne_top (summable_phase_failure_bound_nat m hm).tsum_ofReal_ne_top | |
| (ENNReal.tsum_le_tsum hbad) | |
| end Invariant233.PhaseFailureAudit | |
| end OAI | |