File size: 4,305 Bytes
f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
|