| import OAI.Analysis.FiniteFactor.CrossedProduct.StageOperator |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped ENNReal Topology |
| noncomputable section |
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| theorem sourceCutoff_ge_shift (s n : ℕ) : |
| 2^n * (sourceStageState s).K ≤ (sourceStageState (s+n)).K := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have hh := sourceStage_cutoff_growth (s+n) |
| rw [pow_succ, Nat.add_succ] |
| nlinarith |
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| / |
| theorem sourceCutoff_tail_bound (s : ℕ) : |
| (∑' n : ℕ, 1 / ((sourceStageState (s+n)).K : ℝ)) ≤ |
| 2 / ((sourceStageState s).K : ℝ) := by |
| let b (n : ℕ) : ℝ := (1 / ((sourceStageState s).K : ℝ)) * (1/2 : ℝ)^n |
| have hb : Summable b := (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1/2) |
| (by norm_num : (1/2 : ℝ) < 1)).mul_left _ |
| have hle (n : ℕ) : 1 / ((sourceStageState (s+n)).K : ℝ) ≤ b n := by |
| have hh : (2 : ℝ)^n * (sourceStageState s).K ≤ (sourceStageState (s+n)).K := |
| by exact_mod_cast sourceCutoff_ge_shift s n |
| have hp : (0 : ℝ) < 2^n * (sourceStageState s).K := mul_pos (by positivity) (sourceCutoff_pos s) |
| have ht := one_div_le_one_div_of_le hp hh |
| simpa [b, one_div_pow, mul_comm, mul_left_comm] using ht |
| calc |
| _ ≤ ∑' n, b n := (hb.of_nonneg_of_le (fun _ => by positivity) hle).tsum_le_tsum hle hb |
| _ = 2 / ((sourceStageState s).K : ℝ) := by |
| rw [tsum_mul_left, tsum_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1/2) (by norm_num)] |
| ring |
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| def sourceTailAngleSum (θ : SourceCoordinate → ℝ) (s : ℕ) : ℝ := |
| ∑' n : ℕ, ∑ i : Fin (sourceBlockSize (s+n)), θ ⟨s+n,i⟩ |
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| theorem SourceAngles.tail_sum_bound {θ : SourceCoordinate → ℝ} (hθ : SourceAngles θ) (s : ℕ) : |
| sourceTailAngleSum θ s ≤ 4 / ((sourceStageState s).K : ℝ) := by |
| have hb (n : ℕ) : |
| (∑ i : Fin (sourceBlockSize (s+n)), θ ⟨s+n,i⟩) ≤ 2 / ((sourceStageState (s+n)).K : ℝ) := by |
| have hh := Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => (hθ ⟨s+n,i⟩).2) |
| have he : (∑ _i : Fin (sourceBlockSize (s+n)), |
| 2 / ((sourceBlockSize (s+n) : ℝ)*(sourceStageState (s+n)).K)) = |
| 2 / ((sourceStageState (s+n)).K : ℝ) := by |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| field_simp [(sourceBlockSize_pos (s+n)).ne', (sourceCutoff_pos (s+n)).ne'] |
| exact hh.trans_eq he |
| have hs : Summable (fun n => 2 / ((sourceStageState (s+n)).K : ℝ)) := by |
| have hh := sourceCutoff_reciprocal_summable.comp_injective (add_right_injective s) |
| simpa [div_eq_mul_inv] using hh.mul_left 2 |
| calc |
| sourceTailAngleSum θ s ≤ ∑' n, 2 / ((sourceStageState (s+n)).K : ℝ) := |
| (hs.of_nonneg_of_le (fun n => Finset.sum_nonneg (fun i _ => hθ.nonneg _)) hb).tsum_le_tsum hb hs |
| _ = 2 * ∑' n, 1 / ((sourceStageState (s+n)).K : ℝ) := by |
| rw [← tsum_mul_left]; congr; funext n; ring |
| _ ≤ 2 * (2 / ((sourceStageState s).K : ℝ)) := |
| mul_le_mul_of_nonneg_left (sourceCutoff_tail_bound s) (by norm_num) |
| _ = 4 / ((sourceStageState s).K : ℝ) := by ring |
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| / |
| source exponent gap 7/10 - 3/4 = -1/20. -/ |
| theorem sourceSampling_tail_ratio (s : ℕ) : |
| 4 * (sourceBlockSize s : ℝ)^((7/10 : ℝ)*(sourceBlockSize s : ℝ)) / |
| ((sourceStageState (s+1)).K : ℝ) ≤ |
| 4 * (sourceBlockSize s : ℝ)^(-(sourceBlockSize s : ℝ)/20) := by |
| have hm := sourceBlockSize_pos s |
| have hK := sourceCutoff_pos (s+1) |
| have hc : (sourceBlockSize s : ℝ)^((3/4 : ℝ)*(sourceBlockSize s : ℝ)) ≤ |
| (sourceStageState (s+1)).K := Nat.le_ceil _ |
| apply (div_le_iff₀ hK).mpr |
| calc |
| _ = (4 * (sourceBlockSize s : ℝ)^(-(sourceBlockSize s : ℝ)/20)) * |
| (sourceBlockSize s : ℝ)^((3/4 : ℝ)*(sourceBlockSize s : ℝ)) := by |
| rw [mul_assoc, ← Real.rpow_add hm] |
| congr 2 |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left hc (by positivity) |
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| end |
| end FiniteFactor |
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| end OAI |
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