openAI-math-dataset / lean /OAI /Probability /LogConcave /Dynamics /ObservableIntegralMeasurable.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Dynamics.StationaryPath | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Set MeasureTheory Quadrature RMSIntegral | |
| open scoped Classical BigOperators NNReal | |
| variable {E H : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| [MeasurableSpace E] [BorelSpace E] | |
| [NormedAddCommGroup H] [InnerProductSpace ℝ H] | |
| [MeasurableSpace H] [BorelSpace H] [SecondCountableTopology H] | |
| {μ : Measure E} {V : E → E} | |
| lemma StationaryPath.observableIntegral_measurable (X : StationaryPath μ V) | |
| {M : E → H} (hM : Continuous M) : | |
| Measurable (fun y => ∫t in (0:ℝ)..1,M (X.path y t)) := by | |
| simp only [intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1)] | |
| exact (hM.measurable.comp X.measurable).stronglyMeasurable.integral_prod_left'.measurable | |
| theorem stationaryOutput_rms (X : StationaryPath μ V) | |
| {M : E → H} {L : ℝ≥0} (hM : LipschitzWith L M) | |
| (n : ℕ) (Z : Fin (n+1) → E → E) (hmZ : ∀j,Measurable (Z j)) | |
| {Bz Bq : ℝ} (hBz : 0 ≤ Bz) | |
| (hZi : ∀j,Integrable (fun y => ‖Z j y-X.path y (probabilityNodes n j)‖^2) μ) | |
| (hZb : ∀j,(∫y,‖Z j y-X.path y (probabilityNodes n j)‖^2 ∂μ) ≤ Bz) | |
| (hqi : Integrable (fun y => ‖(∫t in (0:ℝ)..1,M (X.path y t))- | |
| ∑j,weight (probabilityNodes n) j 0 1 • M (X.path y (probabilityNodes n j))‖^2) μ) | |
| (hqb : (∫y,‖(∫t in (0:ℝ)..1,M (X.path y t))- | |
| ∑j,weight (probabilityNodes n) j 0 1 • M (X.path y (probabilityNodes n j))‖^2 ∂μ) ≤ Bq) | |
| (anchor : E → H) : | |
| Integrable (fun y => ‖(anchor y+∑j,weight (probabilityNodes n) j 0 1 • M (Z j y))- | |
| (anchor y+∫t in (0:ℝ)..1,M (X.path y t))‖^2) μ ∧ | |
| (∫y,‖(anchor y+∑j,weight (probabilityNodes n) j 0 1 • M (Z j y))- | |
| (anchor y+∫t in (0:ℝ)..1,M (X.path y t))‖^2 ∂μ) ≤ | |
| 2*(centeringRowBudget n:ℝ)^2*(L:ℝ)^2*Bz+2*Bq := by | |
| let w := fun j => weight (probabilityNodes n) j 0 1 | |
| let e := fun j y => M (Z j y)-M (X.path y (probabilityNodes n j)) | |
| have hem (j : Fin (n+1)) : Measurable (e j) := | |
| (hM.continuous.measurable.comp (hmZ j)).sub | |
| (hM.continuous.measurable.comp (X.measurable_slice _)) | |
| have heb (j : Fin (n+1)) := lipschitz_error_sq (hmZ j) (X.measurable_slice _) | |
| hM (hZi j) (hZb j) | |
| have hsum := weighted_sum_sq w e (fun j => (hem j).aestronglyMeasurable) | |
| (fun j => (heb j).1) (fun j => (heb j).2) | |
| let q := fun y => (∫t in (0:ℝ)..1,M (X.path y t))- | |
| ∑j,w j • M (X.path y (probabilityNodes n j)) | |
| have hqm : Measurable q := (X.observableIntegral_measurable hM.continuous).sub | |
| (Finset.measurable_sum _ (fun j _ => | |
| (hM.continuous.measurable.comp (X.measurable_slice _)).const_smul _)) | |
| have hsm : Measurable (fun y => ∑j,w j • e j y) := | |
| Finset.measurable_sum _ (fun j _ => (hem j).const_smul _) | |
| have hh := sub_sq_bound hsm.aestronglyMeasurable hqm.aestronglyMeasurable | |
| hsum.1 hqi hsum.2 hqb | |
| have hid (y : E) : (anchor y+∑j,w j • M (Z j y))- | |
| (anchor y+∫t in (0:ℝ)..1,M (X.path y t))=(∑j,w j • e j y)-q y := by | |
| simp only [e,q,smul_sub,Finset.sum_sub_distrib] | |
| abel | |
| dsimp only [w] at hid | |
| simp_rw [hid] | |
| refine ⟨hh.1,hh.2.trans ?_⟩ | |
| have hw := pow_le_pow_left₀ (Finset.sum_nonneg (fun _ _ => abs_nonneg _)) | |
| (centeringFullWeight_bound n) 2 | |
| have hv := mul_le_mul_of_nonneg_right hw (show 0 ≤ (L:ℝ)^2*Bz by positivity) | |
| change (∑j,|w j|)^2*((L:ℝ)^2*Bz) ≤ _ at hv | |
| nlinarith | |
| end LogConcaveSampling | |
| end | |
| end | |
| end OAI | |