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b224933 | 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.MeasureTheory.FalconerCompletion.Recursion.MaskScale
import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereAmbient2
namespace OAI
/-! Recursive policies and quantitative depth estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4PrefixShell
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 Attempt4PrefixShell
end GraphLiteralSupport
end
end
end OAI
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