openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /PhaseFailureSummability.lean
| import Mathlib | |
| import OAI.Analysis.IrrationalRotation.Model.PhaseFailure | |
| namespace OAI | |
| open Filter MeasureTheory | |
| open scoped ENNReal Topology | |
| namespace Invariant233.ProductPhases | |
| theorem failureBound_nonneg {m : ℝ} (hm : 0 ≤ m) : 0 ≤ failureBound m := by | |
| unfold failureBound | |
| positivity | |
| theorem ten_mul_phaseWidth_le_one {m : ℝ} (hm : 100 ≤ m) : | |
| 10 * m ^ (-19 / 20 : ℝ) ≤ 1 := by | |
| have hm0 : 0 ≤ m := by linarith | |
| have hs : (10 : ℝ) ≤ Real.sqrt m := by | |
| nlinarith [Real.sq_sqrt hm0, Real.sqrt_nonneg m] | |
| have hb : m ^ (-19 / 20 : ℝ) ≤ (10 : ℝ)⁻¹ := by | |
| calc | |
| m ^ (-19 / 20 : ℝ) ≤ m ^ (-(1 / 2) : ℝ) := | |
| Real.rpow_le_rpow_of_exponent_le (by linarith) (by norm_num) | |
| _ = (Real.sqrt m)⁻¹ := by rw [Real.rpow_neg hm0, ← Real.sqrt_eq_rpow] | |
| _ ≤ (10 : ℝ)⁻¹ := inv_anti₀ (by norm_num) hs | |
| nlinarith | |
| theorem failureBound_eq_exp {m : ℝ} (hm : 0 < m) : | |
| failureBound m = 2 * Real.exp (-(41 / 400 : ℝ) * m * Real.log m + | |
| (Real.log 2 + (19 / 20 : ℝ) * Real.log 10) * m) := by | |
| unfold failureBound | |
| rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 10) (Real.rpow_nonneg hm.le _), | |
| ← Real.rpow_mul hm.le] | |
| simp only [Real.rpow_def_of_pos hm, | |
| Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2), | |
| Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 10)] | |
| rw [mul_assoc, mul_assoc, ← Real.exp_add, ← Real.exp_add, ← Real.exp_add] | |
| congr 2 | |
| ring | |
| theorem eventually_failureBound_le_exp : | |
| ∀ᶠ m : ℝ in atTop, failureBound m ≤ Real.exp (-m) := by | |
| let C : ℝ := Real.log 2 + (19 / 20 : ℝ) * Real.log 10 | |
| have hl : ∀ᶠ m : ℝ in atTop, | |
| (400 / 41 : ℝ) * (C + 2) ≤ Real.log m := | |
| Real.tendsto_log_atTop.eventually (eventually_ge_atTop _) | |
| filter_upwards [eventually_ge_atTop (1 : ℝ), eventually_ge_atTop (Real.log 2), hl] | |
| with m hm hm2 hlog | |
| rw [failureBound_eq_exp (by linarith)] | |
| have he : -(41 / 400 : ℝ) * m * Real.log m + C * m ≤ -2 * m := by | |
| nlinarith [mul_nonneg (sub_nonneg.mpr hlog) (show 0 ≤ m by linarith)] | |
| calc | |
| 2 * Real.exp (-(41 / 400 : ℝ) * m * Real.log m + C * m) = | |
| Real.exp (Real.log 2 + (-(41 / 400 : ℝ) * m * Real.log m + C * m)) := by | |
| conv_rhs => rw [Real.exp_add, Real.exp_log (by norm_num : (0 : ℝ) < 2)] | |
| _ ≤ Real.exp (-m) := Real.exp_le_exp.mpr (by linarith) | |
| theorem summable_failureBound_of_nat_le {m : ℕ → ℝ} | |
| (hm : ∀ j : ℕ, (j : ℝ) ≤ m j) (hmnonneg : ∀ j, 0 ≤ m j) : | |
| Summable (fun j => failureBound (m j)) := by | |
| have hmtop : Tendsto m atTop atTop := | |
| tendsto_atTop_mono hm tendsto_natCast_atTop_atTop | |
| have hb := hmtop.eventually eventually_failureBound_le_exp | |
| apply Real.summable_exp_neg_nat.of_norm_bounded_eventually_nat | |
| filter_upwards [hb] with j hj | |
| rw [Real.norm_eq_abs, abs_of_nonneg (failureBound_nonneg (hmnonneg j))] | |
| exact hj.trans (Real.exp_le_exp.mpr (neg_le_neg (hm j))) | |
| theorem summable_failureBound_of_two_pow_le {m : ℕ → ℝ} | |
| (hm : ∀ j, (2 : ℝ) ^ j ≤ m j) : | |
| Summable (fun j => failureBound (m j)) := by | |
| apply summable_failureBound_of_nat_le | |
| · intro j | |
| exact le_trans (by exact_mod_cast (show j < 2 ^ j from Nat.lt_two_pow_self).le) (hm j) | |
| · intro j | |
| exact le_trans (by positivity) (hm j) | |
| theorem ae_eventually_notMem_of_phase_bound {Ω : Type*} [MeasurableSpace Ω] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (bad : ℕ → Set Ω) {m : ℕ → ℝ} | |
| (hm : ∀ j, (2 : ℝ) ^ j ≤ m j) | |
| (hbad : ∀ᶠ j in atTop, μ (bad j) ≤ ENNReal.ofReal (failureBound (m j))) : | |
| ∀ᵐ ω ∂μ, ∀ᶠ j in atTop, ω ∉ bad j := by | |
| have hs := summable_failureBound_of_two_pow_le hm | |
| have hreal : Summable (fun j => (μ (bad j)).toReal) := by | |
| apply hs.of_norm_bounded_eventually_nat | |
| filter_upwards [hbad] with j hj | |
| rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg] | |
| have hnonneg : 0 ≤ failureBound (m j) := | |
| failureBound_nonneg (le_trans (by positivity) (hm j)) | |
| have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hj | |
| simpa only [ENNReal.toReal_ofReal hnonneg] using hh | |
| apply MeasureTheory.ae_eventually_notMem | |
| have ht := hreal.tsum_ofReal_ne_top | |
| simpa only [ENNReal.ofReal_toReal (measure_ne_top μ _)] using ht | |
| end Invariant233.ProductPhases | |
| end OAI | |