| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.LawRoot | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND | |
| open Set MeasureTheory Filter PlanarFalconer | |
| open scoped ContDiff Classical Topology ENNReal NNReal | |
| variable {n : ℕ} | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| namespace AssemblyClassFamily | |
| variable {ι : Type*} {μ : Measure (EuclideanSpace ℝ (Fin (n+1)))} {N B : ℕ} {chi sig : ℝ} | |
| (F : AssemblyClassFamily ι μ N B chi sig) | |
| lemma law_probability [IsFiniteMeasure μ] (i : ι) : | |
| IsProbabilityMeasure (assemblyClassLaw μ (F.domain i)) := by | |
| constructor | |
| rw [assemblyClassLaw,Measure.smul_apply,smul_eq_mul,Measure.restrict_apply MeasurableSet.univ, | |
| univ_inter] | |
| exact ENNReal.inv_mul_cancel (F.positive i) (measure_ne_top _ _) | |
| end AssemblyClassFamily | |
| section Mixture | |
| variable {ι κ : Type*} [Fintype ι] [Fintype κ] | |
| variable {μ ν : Measure (EuclideanSpace ℝ (Fin (n+1)))} [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| variable {N B : ℕ} {chi sig : ℝ} | |
| variable (F : AssemblyClassFamily ι μ N B chi sig) (G : AssemblyClassFamily κ ν N B chi sig) | |
| variable (kap Ctest beta : ℝ) | |
| noncomputable def assemblyPairCoefficient (a : ι×κ) : ℝ≥0 := F.coefficient a.1 * G.coefficient a.2 | |
| omit [Fintype ι] [Fintype κ] in | |
| lemma assemblyPairCoefficient_prod (a : ι×κ) : | |
| assemblyPairCoefficient F G a • ((F.env kap Ctest a.1).ρ.prod (G.env kap Ctest a.2).ρ) = | |
| (μ.prod ν).restrict (F.domain a.1 ×ˢ G.domain a.2) := by | |
| let := F.law_probability a.1 | |
| let := G.law_probability a.2 | |
| change _ • (assemblyClassLaw μ (F.domain a.1)).prod (assemblyClassLaw ν (G.domain a.2)) = _ | |
| rw [← Measure.prod_restrict,← F.coefficient_law a.1,← G.coefficient_law a.2, | |
| Measure.prod_smul_left,Measure.prod_smul_right,smul_smul] | |
| rfl | |
| omit [Fintype ι] [Fintype κ] in | |
| lemma assemblyPairCoefficient_component (a : ι×κ) (X Y : Set V) | |
| (hX : MeasurableSet X) (hY : MeasurableSet Y) : | |
| assemblyPairCoefficient F G a • assemblyComponentMeasure (F.env kap Ctest a.1) | |
| (G.env kap Ctest a.2) (assemblyPairEntry F G kap Ctest beta a.1 a.2) X Y = | |
| ((μ.prod ν).withDensity ((F.domain a.1 ×ˢ G.domain a.2).indicator | |
| ((X ×ˢ Y).indicator (fun z => ENNReal.ofReal (assemblySpatialWeight | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) z))))).map | |
| (fun z => dist z.1 z.2) := by | |
| let : IsProbabilityMeasure (F.env kap Ctest a.1).ρ := F.law_probability a.1 | |
| let : IsProbabilityMeasure (G.env kap Ctest a.2).ρ := G.law_probability a.2 | |
| rw [assemblyComponentMeasure,← Measure.map_smul _ (by fun_prop)] | |
| congr 1 | |
| rw [Measure.prod_restrict,ENNReal.smul_def,← withDensity_smul_measure, | |
| ← Measure.restrict_smul,← ENNReal.smul_def,assemblyPairCoefficient_prod] | |
| rw [← withDensity_indicator (hX.prod hY), | |
| ← withDensity_indicator ((F.measurable a.1).prod (G.measurable a.2))] | |
| omit [IsFiniteMeasure μ] [IsFiniteMeasure ν] in | |
| lemma assembly_density_eq (I : Finset ℕ) (X Y : ℕ → Set V) | |
| (hdis : Set.Pairwise (↑I : Set ℕ) (fun l m => Disjoint (X l ×ˢ Y l) (X m ×ˢ Y m))) | |
| (z : V×V) : | |
| ENNReal.ofReal ((dist z.1 z.2)^(-(n:ℝ)/2) * | |
| (assemblyGamma I X Y).indicator (assemblyTheta F G kap Ctest beta) z) = | |
| ∑ a : ι×κ, ∑ l∈I, (F.domain a.1 ×ˢ G.domain a.2).indicator | |
| ((X l ×ˢ Y l).indicator (fun z => ENNReal.ofReal (assemblySpatialWeight | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) z))) z := by | |
| have hlocal (S : Set (V×V)) : | |
| (dist z.1 z.2)^(-(n:ℝ)/2) * S.indicator (assemblyTheta F G kap Ctest beta) z = | |
| ∑ a : ι×κ, (F.domain a.1 ×ˢ G.domain a.2).indicator | |
| (S.indicator (assemblySpatialWeight (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2))) z := by | |
| by_cases hs : z∈S | |
| · rw [indicator_of_mem hs] | |
| change (dist z.1 z.2)^(-(n:ℝ)/2) * (∑ i, ∑ j, _) = _ | |
| rw [Finset.mul_sum,Fintype.sum_prod_type] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| by_cases hd : z∈F.domain i ×ˢ G.domain j | |
| · simp only [indicator_of_mem hd,indicator_of_mem hs,assemblySpatialWeight] | |
| · simp only [indicator_of_notMem hd,mul_zero] | |
| · rw [indicator_of_notMem hs,mul_zero] | |
| symm | |
| apply Finset.sum_eq_zero | |
| intro a ha | |
| by_cases hd : z∈F.domain a.1 ×ˢ G.domain a.2 | |
| · simp only [indicator_of_mem hd,indicator_of_notMem hs] | |
| · simp only [indicator_of_notMem hd] | |
| have hreal : (dist z.1 z.2)^(-(n:ℝ)/2) * | |
| (assemblyGamma I X Y).indicator (assemblyTheta F G kap Ctest beta) z = | |
| ∑ a : ι×κ, ∑ l∈I, (F.domain a.1 ×ˢ G.domain a.2).indicator | |
| ((X l ×ˢ Y l).indicator (assemblySpatialWeight | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2))) z := by | |
| rw [assemblyGamma,Finset.indicator_biUnion_apply I (fun l => X l ×ˢ Y l) hdis,Finset.mul_sum] | |
| simp_rw [hlocal] | |
| exact Finset.sum_comm | |
| rw [hreal,ENNReal.ofReal_sum_of_nonneg] | |
| · apply Finset.sum_congr rfl | |
| intro a ha | |
| rw [ENNReal.ofReal_sum_of_nonneg] | |
| · apply Finset.sum_congr rfl | |
| intro l hl | |
| by_cases hd : z∈F.domain a.1 ×ˢ G.domain a.2 <;> | |
| by_cases hx : z∈X l ×ˢ Y l <;> simp [hd,hx] | |
| · intro l hl | |
| exact indicator_nonneg (fun z _ => indicator_nonneg (fun z _ => assemblySpatialWeight_nonneg _ _ _ z) z) z | |
| · intro a ha | |
| exact Finset.sum_nonneg (fun l hl => | |
| indicator_nonneg (fun z _ => indicator_nonneg (fun z _ => assemblySpatialWeight_nonneg _ _ _ z) z) z) | |
| lemma assemblyTau_eq_sum (I : Finset ℕ) (X Y : ℕ → Set V) | |
| (hXY : ∀ l∈I, MeasurableSet (X l) ∧ MeasurableSet (Y l)) | |
| (hdis : Set.Pairwise (↑I : Set ℕ) (fun l m => Disjoint (X l ×ˢ Y l) (X m ×ˢ Y m))) : | |
| assemblyTau F G kap Ctest beta I X Y = | |
| ∑ a : ι×κ, assemblyPairCoefficient F G a • | |
| (∑ l : I, assemblyComponentMeasure (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) (X l) (Y l)) := by | |
| simp only [Finset.smul_sum] | |
| have hsum (a : ι×κ) : | |
| (∑ l:I, assemblyPairCoefficient F G a • assemblyComponentMeasure | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) (X l) (Y l)) = | |
| ∑ l∈I, assemblyPairCoefficient F G a • assemblyComponentMeasure | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) (X l) (Y l) := | |
| Finset.sum_coe_sort I (fun l => assemblyPairCoefficient F G a • assemblyComponentMeasure | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) (X l) (Y l)) | |
| simp_rw [hsum] | |
| have hm (a : ι×κ) (l : ℕ) (hl : l∈I) := | |
| assemblyPairCoefficient_component F G kap Ctest beta a (X l) (Y l) (hXY l hl).1 (hXY l hl).2 | |
| have he : (∑ a : ι×κ, ∑ l∈I, assemblyPairCoefficient F G a • | |
| assemblyComponentMeasure (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) (X l) (Y l)) = | |
| ∑ a : ι×κ, ∑ l∈I, ((μ.prod ν).withDensity ((F.domain a.1 ×ˢ G.domain a.2).indicator | |
| ((X l ×ˢ Y l).indicator (fun z => ENNReal.ofReal (assemblySpatialWeight | |
| (F.env kap Ctest a.1) (G.env kap Ctest a.2) | |
| (assemblyPairEntry F G kap Ctest beta a.1 a.2) z))))).map | |
| (fun z => dist z.1 z.2) := by | |
| apply Finset.sum_congr rfl | |
| intro a ha | |
| exact Finset.sum_congr rfl (fun l hl => hm a l hl) | |
| rw [he,assemblyTau] | |
| simp_rw [← Measure.map_finset_sum ((by fun_prop : Measurable (fun z : V×V => dist z.1 z.2)).aemeasurable)] | |
| congr 1 | |
| ext S hS | |
| simp only [withDensity_apply _ hS,Measure.finsetSum_apply] | |
| simp_rw [assembly_density_eq F G kap Ctest beta I X Y hdis] | |
| rw [lintegral_finsetSum] | |
| · apply Finset.sum_congr rfl | |
| intro a ha | |
| exact lintegral_finsetSum I (fun l hl => | |
| ((assemblySpatialWeight_measurable _ _ _).ennreal_ofReal.indicator | |
| ((hXY l hl).1.prod (hXY l hl).2)).indicator ((F.measurable a.1).prod (G.measurable a.2))) | |
| · intro a ha | |
| exact Finset.measurable_sum I (fun l hl => | |
| ((assemblySpatialWeight_measurable _ _ _).ennreal_ofReal.indicator | |
| ((hXY l hl).1.prod (hXY l hl).2)).indicator ((F.measurable a.1).prod (G.measurable a.2))) | |
| end Mixture | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |