| import Mathlib | |
| import OAI.Probability.LogConcave.Numerics.Picard | |
| import OAI.Probability.LogConcave.Sampling.PrimitiveExpectedFieldLipschitz | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling.MeanTree | |
| open MeasureTheory | |
| open scoped Classical BigOperators NNReal | |
| variable {X Z : Type*} [MeasurableSpace X] [PseudoMetricSpace Z] {d : ℕ} | |
| lemma eval_exists_lipschitz {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) | |
| (E : MeanTree X d) (f : Z → X) | |
| (hb : ∃K : ℝ≥0,LipschitzWith K (fun z => base E (f z))) | |
| (hc : centers (fun r b => 0<r ∧ (lam:ℝ)*r^2≤1/4 ∧ | |
| ∃K : ℝ≥0,LipschitzWith K (fun z => b (f z))) E) : | |
| ∃K : ℝ≥0,LipschitzWith K (fun z => E.eval F (f z)) := by | |
| induction E with | node k b hm a r C ih => | |
| choose K hK using fun i => ih i (hc i).1.2.2 (hc i).2 | |
| let L : ℝ≥0 := ∑ i,K i | |
| have hL (i : Fin k) : K i≤L := Finset.single_le_sum (fun j _ => by positivity) (Finset.mem_univ i) | |
| have hM (i : Fin k) := (primitiveExpectedField_lipschitz hF (hc i).1.1 (hc i).1.2.1).comp ((hK i).weaken (hL i)) | |
| have hs := finite_weighted_lipschitz a (fun i z => primitiveExpectedField F ((C i).eval F (f z)) (r i)) hM | |
| obtain ⟨L,hL⟩ := hb | |
| exact ⟨_,hL.add hs⟩ | |
| end LogConcaveSampling.MeanTree | |
| end | |
| end | |
| end OAI | |