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a2ee19a | 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 Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.RealContinuation
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
lemma real_terminal_cap {d : ℕ} (hd : 1 ≤ d) (y z : Space d)
{ε r α : ℝ} (hε : 0 < ε) (hr : 0 < r) (hα : 0 ≤ α) (hαn : α ≤ (d:ℝ)-1) :
((ambientSphereLaw (E := Space d)).restrict (Metric.closedBall y ε)) (Metric.closedBall z r) ≤
ENNReal.ofReal ((3*4^d:ℝ) * ε^((d:ℝ)-1-α) * r^α) := by
let : NeZero d := ⟨by omega⟩
have hn : ((d-1:ℕ):ℝ) = (d:ℝ)-1 := by rw [Nat.cast_sub hd, Nat.cast_one]
have hs : 0 ≤ (d:ℝ)-1-α := sub_nonneg.mpr hαn
have he (t : ℝ) (ht : 0 < t) : t^(d-1) = t^((d:ℝ)-1-α)*t^α := by
rw [← Real.rpow_add ht, sub_add_cancel, ← hn, Real.rpow_natCast]
rw [Measure.restrict_apply Metric.isClosed_closedBall.measurableSet]
by_cases h : r ≤ ε
· calc
_ ≤ ambientSphereLaw (E := Space d) (Metric.closedBall z r) := measure_mono Set.inter_subset_left
_ ≤ ENNReal.ofReal ((3*4^d:ℝ)*r^(d-1)) := by
simpa [ENNReal.ofReal_mul, finrank_euclideanSpace_fin] using
ambientSphereLaw_closedBall (E := Space d) z hr (d-1) (by simp)
_ ≤ _ := by
apply ENNReal.ofReal_le_ofReal
rw [he r hr, ← mul_assoc]
exact mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hr.le h hs) (by positivity)) (by positivity)
· calc
_ ≤ ambientSphereLaw (E := Space d) (Metric.closedBall y ε) := measure_mono Set.inter_subset_right
_ ≤ ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by
simpa [ENNReal.ofReal_mul, finrank_euclideanSpace_fin] using
ambientSphereLaw_closedBall (E := Space d) y hε (d-1) (by simp)
_ ≤ _ := by
apply ENNReal.ofReal_le_ofReal
rw [he ε hε, ← mul_assoc]
exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hε.le (le_of_not_ge h) hα) (by positivity)
lemma sphere_cap_real_mass {d : ℕ} (hd : 1 ≤ d) (y : Space d) {ε α : ℝ}
(hε : 0 < ε) (hα : 0 ≤ α) (hαn : α ≤ (d:ℝ)-1) :
SphereBallBound (fun s => capWeight y ε s.val) ((3*4^d:ℝ)*ε^((d:ℝ)-1-α)) α := by
intro z r hr
change (∫⁻ s in Subtype.val ⁻¹' Metric.closedBall z r,
capWeight y ε s.val ∂sphereLaw) ≤ _
rw [← setLIntegral_map Metric.isClosed_closedBall.measurableSet (measurable_capWeight y ε)
measurable_subtype_coe]
change (∫⁻ s in Metric.closedBall z r, capWeight y ε s ∂ambientSphereLaw) ≤ _
rw [← withDensity_apply _ Metric.isClosed_closedBall.measurableSet]
change ((ambientSphereLaw (E := Space d)).withDensity
((Metric.closedBall y ε).indicator 1)) (Metric.closedBall z r) ≤ _
rw [withDensity_indicator_one Metric.isClosed_closedBall.measurableSet]
exact real_terminal_cap hd y z hε hr hα hαn
lemma sphere_mass_bound {d : ℕ} {h : Sphere d → ℝ≥0∞} {H α : ℝ}
(hh : SphereBallBound h H α) : (∫⁻ s, h s ∂SpherePrior d) ≤ ENNReal.ofReal H := by
have he : {s : Sphere d | dist s.val (0:Space d) ≤ 1} = Set.univ := by
ext s
simp [dist_zero_right]
simpa [SphereBallBound, he] using hh 0 1 zero_lt_one
lemma real_continuation_cap_integral {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N)
{δ : ℕ → Fin N → (Space d × ℝ) → Fin N} (hδ : ∀ n j, Measurable (δ n j))
(y : ℕ → Fin N → Space d) {ε α : ℝ} (hε : 0 < ε)
(hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) (n k : ℕ) (j : Fin N) :
(∫⁻ s, continuation δ (fun n j => capWeight (y n j) ε) n k j s.val ∂SpherePrior d) ≤
ENNReal.ofReal ((3*4^d:ℝ)*ε^((d:ℝ)-1-α) * RealStep d N ^ BlockCount k (d/8)) := by
have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd
apply sphere_mass_bound
apply real_continuation_mass hd hN hδ (fun n j => measurable_capWeight _ _)
(fun n j s => capWeight_le_one _ _ _) (by positivity) hαlo hαhi
(fun n j => sphere_cap_real_mass (by omega) (y n j) hε (by linarith) (by linarith))
simpa [mul_comm] using blockCount_mul_ge k (by omega : 0 < d/8)
end MemoryPrecision
end
end OAI
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