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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 | 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
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