openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Estimates /MinimalMaskNestedSupport.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.MinimalMask | |
| namespace OAI | |
| /-! Quantitative estimates for the distance-set construction. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ENNReal | |
| variable {d n : ℕ} | |
| namespace PositiveDistanceState | |
| lemma uniform_minimalMask_nested_support {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : | |
| ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState d) [IsFiniteMeasure e.ρ], | |
| N₀≤e.N → e.sig=sig → e.J=J → 0<e.q → e.t≤e.N → | |
| ∀ v≤e.t, ∀ z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d), z.1≠z.2 → | |
| e.minimalMask z≠0 → z∈({e with t:=v,i:=e.i+1} : PositiveDistanceState d).good := by | |
| obtain ⟨N₀,h₀⟩ := uniform_actualEndpointMask_support (d:=d) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro e he hN hsig' hJ' hq ht v hv z hz hm b hb | |
| have hb' := policyDescriptorList_mono (adjustedProfile (d:=d) e.mass e.N) (le_refl e.a) hv hb | |
| have hp := Finset.prod_ne_zero_iff.mp hm b hb' | |
| have hvalid := policyDescriptor_valid _ hq ht hb' | |
| have hmask (x : EuclideanSpace ℝ (Fin d)) | |
| (hx : actualEndpointMask e.ρ e.mass e.N e.sig e.C e.J e.i b x (radialVector z.1 z.2)≠0) : | |
| (x,radialVector z.1 z.2)∈endpointGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) b := by | |
| have h := h₀ e.N hN e.ρ he e.mass e.C e.i b hvalid 0 x (radialVector z.1 z.2) | |
| (norm_radialVector hz) | |
| rw [← hsig',← hJ'] at h | |
| apply h | |
| apply subset_tsupport | |
| rw [Function.mem_support,← norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff] | |
| exact hx | |
| exact ⟨hmask z.1 (mul_ne_zero_iff.mp hp).1,hmask z.2 (mul_ne_zero_iff.mp hp).2⟩ | |
| lemma weightedDistance_le_nested (e : PositiveDistanceState (n+1)) [IsFiniteMeasure e.ρ] | |
| {K : ℝ} (hK : 32*(n+2:ℝ)≤K) (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| (hs : farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) | |
| (v : ℕ) (hg : ∀ᵐ z ∂e.cellPair, e.spatialWeight z≠0 → | |
| z∈({e with t:=v,i:=e.i+1} : PositiveDistanceState (n+1)).good) : | |
| e.weightedDistance≤ENNReal.ofReal (e.levelMass^(-2:ℤ)) • | |
| ({e with t:=v,i:=e.i+1} : PositiveDistanceState (n+1)).distance := by | |
| let f : PositiveDistanceState (n+1) := {e with t:=v,i:=e.i+1} | |
| have hp : e.weightedPair≤ENNReal.ofReal (e.levelMass^(-2:ℤ)) • f.pair := by | |
| change e.weightedPair≤ENNReal.ofReal (e.levelMass^(-2:ℤ)) • e.cellPair.restrict f.good | |
| rw [← withDensity_const,← withDensity_indicator f.measurable_good] | |
| apply withDensity_mono | |
| filter_upwards [e.spatialWeight_bounds hK hX hY hs,hg] with z hz hg | |
| by_cases hw : e.spatialWeight z=0 | |
| · simp only [hw,mul_zero,ENNReal.ofReal_zero] | |
| exact zero_le | |
| · rw [Set.indicator_of_mem (hg hw)] | |
| exact ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right (by positivity) hz.1.2) | |
| have h := Measure.map_mono hp (continuous_fst.dist continuous_snd).measurable | |
| rw [Measure.map_smul _ (continuous_fst.dist continuous_snd).measurable.aemeasurable] at h | |
| simpa only [weightedDistance,distance] using h | |
| lemma uniform_weightedDistance_nested_domination {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : | |
| ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)) [IsFiniteMeasure e.ρ], | |
| N₀≤e.N → e.sig=sig → e.J=J → 0<e.q → e.t≤e.N → | |
| ∀ K : ℝ, 32*(n+2:ℝ)≤K → e.X.length=e.a → e.Y.length=e.a → | |
| farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) → | |
| ∀ v≤e.t, e.weightedDistance≤ENNReal.ofReal (e.levelMass^(-2:ℤ)) • | |
| ({e with t:=v,i:=e.i+1} : PositiveDistanceState (n+1)).distance := by | |
| obtain ⟨N₀,h₀⟩ := uniform_minimalMask_nested_support (d:=n+1) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro e he hN hsig' hJ' hq ht K hK hX hY hs v hv | |
| apply e.weightedDistance_le_nested hK hX hY hs v | |
| filter_upwards [e.ae_cellPair_mem] with z hz | |
| intro hw | |
| have hne : z.1≠z.2 := by | |
| have hh := (e.separated_distance_ratio hK hX hY hs hz.1 hz.2).1 | |
| intro he | |
| simp only [he,dist_self,zero_mul] at hh | |
| norm_num at hh | |
| have hm : e.minimalMask z≠0 := by | |
| intro h | |
| exact hw (by simp only [spatialWeight,h,mul_zero]) | |
| exact h₀ e hN hsig' hJ' hq ht v hv z hne hm | |
| end PositiveDistanceState | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |