| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.ProfileChildCard |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.NominalRadius |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace Attempt4FullFar |
| open Filter Real |
| open scoped Topology |
|
|
| noncomputable def sourceMaskScaleND (l : ℕ) (R σ : ℝ) : ℝ := |
| max 1 ((2:ℝ)^l*R^(-50*σ)) |
|
|
| noncomputable def sourceMaskRadiusND (d l : ℕ) (R σ : ℝ) : ℝ := |
| (sourceMaskScaleND l R σ)⁻¹/(100*(d+1)^2) |
|
|
| noncomputable def sourceTestWidthND (J i n : ℕ) (R σ : ℝ) : ℝ := |
| (2-(i:ℝ)/J)*((2:ℝ)^n)⁻¹*R^(500*σ) |
|
|
| lemma sourceMaskScaleND_one_le (l : ℕ) (R σ : ℝ) : 1 ≤ sourceMaskScaleND l R σ := le_max_left _ _ |
|
|
| lemma sourceMaskScaleND_pos (l : ℕ) (R σ : ℝ) : 0<sourceMaskScaleND l R σ := |
| zero_lt_one.trans_le (sourceMaskScaleND_one_le l R σ) |
|
|
| lemma sourceMaskRadiusND_pos (d l : ℕ) (R σ : ℝ) : 0<sourceMaskRadiusND d l R σ := |
| div_pos (inv_pos.mpr (sourceMaskScaleND_pos l R σ)) (by positivity) |
|
|
| lemma sourceMaskScaleND_mono {l L : ℕ} (h : l ≤ L) {R σ : ℝ} (hR : 0 ≤ R) : |
| sourceMaskScaleND l R σ ≤ sourceMaskScaleND L R σ := by |
| apply max_le_max_left |
| exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num) h) (Real.rpow_nonneg hR _) |
|
|
| lemma sourceMaskScaleND_inv_depth {g p : ℕ} (hgp : g ≤ p) {R σ : ℝ} (hR : 0<R) : |
| ((2:ℝ)^g)⁻¹*(sourceMaskScaleND (p-g) R σ)⁻¹ ≤ ((2:ℝ)^p)⁻¹*R^(50*σ) := by |
| have hpos : 0<(2:ℝ)^(p-g)*R^(-50*σ) := by positivity |
| have hh := inv_anti₀ hpos (le_max_right (1:ℝ) ((2:ℝ)^(p-g)*R^(-50*σ))) |
| apply (mul_le_mul_of_nonneg_left hh (by positivity : (0:ℝ) ≤ ((2:ℝ)^g)⁻¹)).trans_eq |
| rw [mul_inv_rev,show -50*σ=-(50*σ) by ring,Real.rpow_neg hR.le,inv_inv] |
| calc |
| _ = (((2:ℝ)^g)*((2:ℝ)^(p-g)))⁻¹*R^(50*σ) := by rw [mul_inv_rev]; ring |
| _ = _ := by rw [← pow_add,Nat.add_sub_of_le hgp] |
|
|
| lemma sourceTestWidthND_gap {J : ℕ} (hJ : 0<J) (i n : ℕ) (R σ : ℝ) : |
| sourceTestWidthND J i n R σ-sourceTestWidthND J (i+1) n R σ = |
| (J:ℝ)⁻¹*((2:ℝ)^n)⁻¹*R^(500*σ) := by |
| unfold sourceTestWidthND |
| push_cast |
| have hJ' : (J:ℝ) ≠ 0 := by exact_mod_cast hJ.ne' |
| field_simp |
| ring |
|
|
| lemma source_mask_width_gapND {d g p J i : ℕ} {R σ : ℝ} |
| (hgp : g ≤ p) (hJ : 0<J) (hR : 0<R) (hlarge : (J:ℝ) ≤ R^(450*σ)) : |
| sourceTestWidthND J (i+1) p R σ+ |
| 40*(d+1)^2/(2:ℝ)^g*sourceMaskRadiusND d (p-g) R σ ≤ |
| sourceTestWidthND J i p R σ := by |
| have hD := sourceMaskScaleND_inv_depth hgp hR (σ := σ) |
| have hJn : (0:ℝ)<J := by exact_mod_cast hJ |
| have hpow : R^(50*σ) ≤ (J:ℝ)⁻¹*R^(500*σ) := by |
| calc |
| _ ≤ ((J:ℝ)⁻¹*R^(450*σ))*R^(50*σ) := by |
| apply le_mul_of_one_le_left (Real.rpow_nonneg hR.le _) |
| exact (one_le_inv_mul₀ hJn).mpr hlarge |
| _ = _ := by rw [mul_assoc,← Real.rpow_add hR]; congr 2; ring |
| have hpert : 40*(d+1)^2/(2:ℝ)^g*sourceMaskRadiusND d (p-g) R σ ≤ |
| (J:ℝ)⁻¹*((2:ℝ)^p)⁻¹*R^(500*σ) := by |
| unfold sourceMaskRadiusND |
| calc |
| _ = (2/5:ℝ)*(((2:ℝ)^g)⁻¹*(sourceMaskScaleND (p-g) R σ)⁻¹) := by field_simp; ring |
| _ ≤ (2/5:ℝ)*(((2:ℝ)^p)⁻¹*R^(50*σ)) := mul_le_mul_of_nonneg_left hD (by norm_num) |
| _ ≤ ((2:ℝ)^p)⁻¹*R^(50*σ) := mul_le_of_le_one_left (by positivity) (by norm_num) |
| _ ≤ _ := by simpa only [mul_left_comm,mul_assoc] using mul_le_mul_of_nonneg_left hpow (by positivity : (0:ℝ) ≤ ((2:ℝ)^p)⁻¹) |
| have he := sourceTestWidthND_gap hJ i p R σ |
| linarith |
|
|
| theorem source_mask_width_gap_eventuallyND {σ : ℝ} (hσ : 0<σ) (d J : ℕ) (hJ : 0<J) : |
| ∀ᶠ R : ℝ in atTop, ∀ g p i : ℕ, g ≤ p → |
| sourceTestWidthND J (i+1) p R σ+ |
| 40*(d+1)^2/(2:ℝ)^g*sourceMaskRadiusND d (p-g) R σ ≤ |
| sourceTestWidthND J i p R σ := by |
| filter_upwards [eventually_gt_atTop (0:ℝ), |
| (tendsto_rpow_atTop (by positivity : (0:ℝ)<450*σ)).eventually_ge_atTop (J:ℝ)] with R hR hlarge |
| intro g p i hgp |
| exact source_mask_width_gapND hgp hJ hR hlarge |
|
|
| end Attempt4FullFar |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
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|
