openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Geometry /ConditionalProjectionBuffer.lean
| import OAI.MeasureTheory.Falconer.Geometry.ProjectionBuffer | |
| import OAI.MeasureTheory.Falconer.Estimates.RescaledConditional | |
| namespace OAI | |
| /-! # Projection buffers for the full original conditional laws -/ | |
| namespace Falconer.Regularization | |
| open Set MeasureTheory | |
| variable {d : ℕ} | |
| theorem conditional_pair_diameter (Q : DyadicCube d) (hQ : 0 < Q.side) | |
| (ρ : Measure (Space d)) [IsFiniteMeasure ρ] (P : CellPath d) : | |
| ∀ᵐ z : Space d × Space d ∂ | |
| (ProbabilityTheory.cond ρ (Q.descendant P).carrier).prod | |
| (ProbabilityTheory.cond ρ (Q.descendant P).carrier), | |
| ‖z.1-z.2‖ ≤ Real.sqrt d*(Q.side/(2:ℝ)^P.length) := by | |
| have hs : ∀ᵐ z ∂ProbabilityTheory.cond ρ (Q.descendant P).carrier, | |
| z ∈ (Q.descendant P).carrier := | |
| ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant P).measurable (fun _ hz => hz) | |
| apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr | |
| filter_upwards [hs] with x hx | |
| filter_upwards [hs] with y hy | |
| have hh := (Q.descendant P).dist_le (by rw [Q.descendant_side]; positivity) hx hy | |
| simpa only [dist_eq_norm,Q.descendant_side] using hh | |
| theorem conditional_projection_buffer (Q : DyadicCube d) (hQ : Q.side = 1) | |
| (ρ : Measure (Space d)) [IsFiniteMeasure ρ] (P : CellPath d) | |
| {δ W W' T : ℝ} (hδ : 0 ≤ δ) {v w : Sphere d} (hvw : dist v w ≤ δ) | |
| (hwidth : W+(Real.sqrt d/(2:ℝ)^P.length)*δ ≤ W') | |
| (hw : w ∈ projectionGoodSet (ProbabilityTheory.cond ρ (Q.descendant P).carrier) W' T) : | |
| v ∈ projectionGoodSet (ProbabilityTheory.cond ρ (Q.descendant P).carrier) W T := by | |
| apply projectionGoodSet_buffer _ (by positivity) hδ _ hvw hwidth hw | |
| simpa only [hQ,mul_one_div] using conditional_pair_diameter Q (by rw [hQ]; norm_num) ρ P | |
| end Falconer.Regularization | |
| end OAI | |