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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 88 89 90 91 | import OAI.Analysis.IrrationalRotation.Sampling.CirclePhaseSummability
namespace OAI
noncomputable section
open Filter
open scoped BigOperators Topology
namespace Invariant233.CircleParameterLaws
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
theorem block_phase_tail_bound (j : ℕ) (hj : 1 ≤ j) :
(∑' n : ℕ, ∑ i ∈ p.block (j + 1 + n), p.alpha θ i) ≤
4 / (p.K (j + 1) : ℝ) := by
have hs : Summable (fun n : ℕ =>
∑ i ∈ p.block (j + 1 + n), p.alpha θ i) := by
have h := (summable_nat_add_iff j).mpr hp.summable_block_phases
simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using h
have hK : Summable (fun n : ℕ => 1 / (p.K (j + 1 + n) : ℝ)) := by
have h := (summable_nat_add_iff j).mpr hp.summable_reciprocal
simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using h
calc
_ ≤ ∑' n : ℕ, 2 * (1 / (p.K (j + 1 + n) : ℝ)) := by
apply Summable.tsum_le_tsum _ hs (hK.mul_left 2)
intro n
simpa only [mul_one_div] using hp.block_phase_sum_le (j + 1 + n) (by omega)
_ = 2 * ∑' n : ℕ, 1 / (p.K (j + 1 + n) : ℝ) := tsum_mul_left
_ ≤ 2 * (2 / (p.K (j + 1) : ℝ)) :=
mul_le_mul_of_nonneg_left (hp.reciprocal_tail_bound j hj) (by norm_num)
_ = _ := by ring
theorem sampled_phase_tail_bound (j r n : ℕ) (hj : 1 ≤ j)
(hr : r < p.q (p.blockEnd (j - 1))) (hn : n ∈ p.samples j r) :
(n : ℝ) * (∑' s : ℕ, ∑ i ∈ p.block (j + 1 + s), p.alpha θ i) ≤
4 * p.Q j / (p.K (j + 1) : ℝ) := by
have hK : 0 < (p.K (j + 1) : ℝ) := by
exact_mod_cast hp.K_pos (j + 1) (by omega)
calc
_ ≤ (n : ℝ) * (4 / (p.K (j + 1) : ℝ)) :=
mul_le_mul_of_nonneg_left (hp.block_phase_tail_bound j hj) (Nat.cast_nonneg n)
_ ≤ p.Q j * (4 / (p.K (j + 1) : ℝ)) :=
mul_le_mul_of_nonneg_right (hp.samples_range j hj r hr n hn).2 (by positivity)
_ = _ := by ring
theorem early_trimming_bound (j : ℕ) :
(p.blockEnd (j - 1) : ℝ) * (p.q (p.blockEnd (j - 1)) : ℝ) * p.ell j ≤
1 / ((j + 1 : ℕ) : ℝ)^2 := by
have hq : 0 < (p.q (p.blockEnd (j - 1)) : ℝ) := by
exact_mod_cast hp.q_pos (p.blockEnd (j - 1))
have ha : 0 ≤ (p.blockEnd (j - 1) : ℝ) := Nat.cast_nonneg _
have hr : (p.blockEnd (j - 1) : ℝ) / (1 + (p.blockEnd (j - 1) : ℝ)) ≤ 1 := by
apply (div_le_one (by positivity)).mpr
linarith
calc
_ = ((p.blockEnd (j - 1) : ℝ) / (1 + (p.blockEnd (j - 1) : ℝ))) *
(1 / ((j + 1 : ℕ) : ℝ)^2) := by
unfold CircleParameterData.ell
field_simp
_ ≤ 1 * (1 / ((j + 1 : ℕ) : ℝ)^2) :=
mul_le_mul_of_nonneg_right hr (by positivity)
_ = _ := one_mul _
theorem discarded_mass_bound (j : ℕ) (hj : 1 ≤ j) :
(p.blockEnd (j - 1) : ℝ) * (p.q (p.blockEnd (j - 1)) : ℝ) * p.ell j +
2 * p.Q j * (∑' n : ℕ, 1 / (p.K (j + 1 + n) : ℝ)) ≤
1 / ((j + 1 : ℕ) : ℝ)^2 + 4 * p.Q j / (p.K (j + 1) : ℝ) := by
apply add_le_add (hp.early_trimming_bound j)
have hQ : 0 ≤ p.Q j := Real.rpow_nonneg (Nat.cast_nonneg _) _
calc
_ ≤ 2 * p.Q j * (2 / (p.K (j + 1) : ℝ)) :=
mul_le_mul_of_nonneg_left (hp.reciprocal_tail_bound j hj) (by positivity)
_ = _ := by ring
theorem tendsto_discarded_mass_bound :
Tendsto (fun n : ℕ => 1 / ((n + 2 : ℕ) : ℝ)^2 +
4 * p.Q (n + 1) / (p.K (n + 2) : ℝ)) atTop (𝓝 0) := by
have h : Tendsto (fun n : ℕ => 1 / ((n + 2 : ℕ) : ℝ)) atTop (𝓝 (0 : ℝ)) := by
exact (tendsto_add_atTop_iff_nat 2).mpr tendsto_one_div_atTop_nhds_zero_nat
have hsq : Tendsto (fun n : ℕ => 1 / ((n + 2 : ℕ) : ℝ)^2)
atTop (𝓝 (0 : ℝ)) := by
simpa only [one_div_pow, zero_pow (by decide : 2 ≠ 0)] using h.pow 2
have hg := hsq.add (hp.tendsto_sampling_horizon.const_mul 4)
simpa only [mul_div_assoc, zero_add, mul_zero] using hg
end Invariant233.CircleParameterLaws
end
end OAI
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