openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphChildrenHSplit.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.Ledger | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AngularHCongr | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFixedAngularBudget | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory Attempt4Near Attempt4FullFar | |
| open scoped ComplexConjugate RealInnerProductSpace ContDiff | |
| variable {d : ℕ} | |
| lemma graph_children_H_split | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| {a p : ℕ} (hap : a ≤ p) {X Y : CellPath d} (hX : X ∈ dyadicPaths a) (hY : Y ∈ dyadicPaths a) | |
| {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ} | |
| {m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B) | |
| (Ma₁ Ma₂ : ℝ) {Mp₁ Mp₂ : ℝ} (hMp₁ : Mp₁ ≠ 0) (hMp₂ : Mp₂ ≠ 0) (v : ℝ) (Ψ : ℝ → ℝ) : | |
| sourceAngularHND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ | |
| (fun z => sourceAngularTransformND ρ (treeCell X) Ma₁ b z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell Y) Ma₂ c z.1 z.2) = | |
| sourceAngularHND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ | |
| (fun z => (Mp₁/Ma₁) • ∑ P ∈ graphChildren ρ a p X, sourceAngularTransformND ρ (treeCell P) Mp₁ b z.1 z.2) | |
| (fun z => (Mp₂/Ma₂) • ∑ Q ∈ graphChildren κ a p Y, sourceAngularTransformND κ (treeCell Q) Mp₂ c z.1 z.2) := by | |
| apply sourceAngularHND_congr | |
| · intro r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact graph_children_transform_split ρ hap hX hb hMp₁ r w hw | |
| · intro r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact graph_children_transform_split κ hap hY hc hMp₂ r w hw | |
| lemma graph_children_HF_split | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| {a p : ℕ} (hap : a ≤ p) {X Y : CellPath d} (hX : X ∈ dyadicPaths a) (hY : Y ∈ dyadicPaths a) | |
| {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ} | |
| {m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B) | |
| (Ma₁ Ma₂ : ℝ) {Mp₁ Mp₂ : ℝ} (hMp₁ : Mp₁ ≠ 0) (hMp₂ : Mp₂ ≠ 0) | |
| (v : ℝ) (Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ) : | |
| sourceAngularHFND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ ψ | |
| (fun z => sourceAngularTransformND ρ (treeCell X) Ma₁ b z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell Y) Ma₂ c z.1 z.2) = | |
| sourceAngularHFND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ ψ | |
| (fun z => (Mp₁/Ma₁) • ∑ P ∈ graphChildren ρ a p X, sourceAngularTransformND ρ (treeCell P) Mp₁ b z.1 z.2) | |
| (fun z => (Mp₂/Ma₂) • ∑ Q ∈ graphChildren κ a p Y, sourceAngularTransformND κ (treeCell Q) Mp₂ c z.1 z.2) := by | |
| apply sourceAngularHFND_congr | |
| · intro r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact graph_children_transform_split ρ hap hX hb hMp₁ r w hw | |
| · intro r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact graph_children_transform_split κ hap hY hc hMp₂ r w hw | |
| lemma graph_direct_product_identity (z : ℂ) (e₁ e₂ : ℝ) (u u' v v' : ℂ) : | |
| z*((e₁:ℂ)*u*conj u')*((e₂:ℂ)*v*conj v') = | |
| z*((e₁:ℂ)*(e₂:ℂ))*((u*v)*conj (u'*v')) := by | |
| simp only [map_mul] | |
| ring | |
| lemma graph_crossed_product_identity (z : ℂ) (e₁ e₂ : ℝ) (s t u u' v v' : ℂ) : | |
| z*((e₁:ℂ)*s*u*conj v')*((e₂:ℂ)*conj t*v*conj u') = | |
| z*((e₁:ℂ)*(e₂:ℂ))*(((s*u)*v)*conj ((t*u')*v')) := by | |
| simp only [map_mul] | |
| ring | |
| lemma graph_direct_cell_integrable | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| (M₁ M₂ : ℝ) {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ} | |
| {m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B) | |
| {Ψ : ℝ → ℝ} (v : ℝ) (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) | |
| (δ : ℝ) (k₁ k₂ : Fin d → ℤ) (row col : CellPath d × CellPath d) : | |
| Integrable (fun q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))) => | |
| (Ψ (q.1/(2:ℝ)^v):ℂ)*((tensorCapND δ k₁ q.2.1*tensorCapND δ k₂ q.2.2):ℂ)* | |
| ((sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1 * | |
| sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2)* | |
| conj (sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.1 * | |
| sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.2))) | |
| (volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod | |
| (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))) := by | |
| have h := graph_angular_four_integrable (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) | |
| hΨ.ofReal (fun w => (tensorCapND δ k₁ w:ℂ)) (fun w => (tensorCapND δ k₂ w:ℂ)) | |
| (Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₁).continuous.measurable) | |
| (Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₂).continuous.measurable) | |
| (fun z => sourceAngularTransformND ρ (treeCell row.1) M₁ b z.1 z.2) | |
| (fun z => sourceAngularTransformND ρ (treeCell col.1) M₁ b z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell row.2) M₂ c z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell col.2) M₂ c z.1 z.2) | |
| (sourceAngularTransformND_measurable _ _ _ hb.measurable) | |
| (sourceAngularTransformND_measurable _ _ _ hb.measurable) | |
| (sourceAngularTransformND_measurable _ _ _ hc.measurable) | |
| (sourceAngularTransformND_measurable _ _ _ hc.measurable) | |
| (P₁:=1) (P₂:=1) | |
| (B₁:=|M₁⁻¹| *ρ.real (treeCell row.1)*B) (B₂:=|M₁⁻¹| *ρ.real (treeCell col.1)*B) | |
| (B₃:=|M₂⁻¹| *κ.real (treeCell row.2)*B) (B₄:=|M₂⁻¹| *κ.real (treeCell col.2)*B) | |
| (by norm_num) (by norm_num) (by positivity [hb.nonneg]) (by positivity [hb.nonneg]) | |
| (by positivity [hc.nonneg]) (by positivity [hc.nonneg]) ?_ | |
| · apply h.congr | |
| filter_upwards with q | |
| exact graph_direct_product_identity _ _ _ _ _ _ _ | |
| · filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| refine ⟨?_,?_,fun r => ⟨?_,?_,?_,?_⟩⟩ | |
| · simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _)] using tensorCapND_le_one δ k₁ w | |
| · simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _)] using tensorCapND_le_one δ k₂ w | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw) | |
| lemma graph_crossed_cell_integrable | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| (M₁ M₂ : ℝ) {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ} | |
| {m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hc : GraphSymbolJet c m s B) | |
| {Ψ : ℝ → ℝ} (v : ℝ) (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) | |
| (ψ : EuclideanSpace ℝ (Fin d) → ℂ) (hmψ : Measurable ψ) | |
| {P : ℝ} (hP : 0 ≤ P) (hψ : ∀ w, ‖w‖=1 → ‖ψ w‖ ≤ P) | |
| (δ : ℝ) (k₁ k₂ : Fin d → ℤ) (row col : CellPath d × CellPath d) : | |
| Integrable (fun q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))) => | |
| (Ψ (q.1/(2:ℝ)^v):ℂ)*((tensorCapND δ k₁ q.2.1*tensorCapND δ k₂ q.2.2):ℂ)* | |
| (((ψ q.2.1*sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1) * | |
| sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2)* | |
| conj ((ψ q.2.2*sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.2) * | |
| sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.1))) | |
| (volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod | |
| (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))) := by | |
| have h := graph_angular_four_integrable (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) | |
| hΨ.ofReal (fun w => (tensorCapND δ k₁ w:ℂ)*ψ w) (fun w => (tensorCapND δ k₂ w:ℂ)*conj (ψ w)) | |
| ((Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₁).continuous.measurable).mul hmψ) | |
| ((Complex.ofRealCLM.continuous.measurable.comp (tensorCapND_contDiff δ k₂).continuous.measurable).mul (Complex.continuous_conj.measurable.comp hmψ)) | |
| (fun z => sourceAngularTransformND ρ (treeCell row.1) M₁ b z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell col.2) M₂ c z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell row.2) M₂ c z.1 z.2) | |
| (fun z => sourceAngularTransformND ρ (treeCell col.1) M₁ b z.1 z.2) | |
| (sourceAngularTransformND_measurable _ _ _ hb.measurable) | |
| (sourceAngularTransformND_measurable _ _ _ hc.measurable) | |
| (sourceAngularTransformND_measurable _ _ _ hc.measurable) | |
| (sourceAngularTransformND_measurable _ _ _ hb.measurable) | |
| (P₁:=P) (P₂:=P) | |
| (B₁:=|M₁⁻¹| *ρ.real (treeCell row.1)*B) (B₂:=|M₂⁻¹| *κ.real (treeCell col.2)*B) | |
| (B₃:=|M₂⁻¹| *κ.real (treeCell row.2)*B) (B₄:=|M₁⁻¹| *ρ.real (treeCell col.1)*B) | |
| hP hP (by positivity [hb.nonneg]) (by positivity [hc.nonneg]) | |
| (by positivity [hc.nonneg]) (by positivity [hb.nonneg]) ?_ | |
| · apply h.congr | |
| filter_upwards with q | |
| exact graph_crossed_product_identity _ _ _ _ _ _ _ _ _ | |
| · filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| refine ⟨?_,?_,fun r => ⟨?_,?_,?_,?_⟩⟩ | |
| · simpa only [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _),one_mul] using | |
| mul_le_mul (tensorCapND_le_one δ k₁ w) (hψ w hw) (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 1) | |
| · simpa only [norm_mul,Complex.norm_conj,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (tensorCapND_nonneg _ _ _),one_mul] using | |
| mul_le_mul (tensorCapND_le_one δ k₂ w) (hψ w hw) (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 1) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw) | |
| · exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw) | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory PlanarFalconer Attempt4FullFar | |
| open scoped ComplexConjugate RealInnerProductSpace ContDiff | |
| variable {d : ℕ} | |
| attribute [local instance] Classical.propDecidable | |
| lemma graphCapScale_pos {N a p : ℕ} {c₀ sig : ℝ} (hc₀ : 0 < c₀) : | |
| 0 < graphCapScale N a p c₀ sig := lt_min hc₀ (by positivity) | |
| lemma graphCapScale_eq (N a p : ℕ) (c₀ sig : ℝ) : | |
| graphCapScale N a p c₀ sig = min c₀ (((2:ℝ)^((p:ℝ)-a))⁻¹*((2:ℝ)^N)^(10*sig)) := by | |
| simp only [graphCapScale,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2)] | |
| lemma graph_pair_partition_ae {N a p : ℕ} {c₀ sig : ℝ} | |
| (hc₀ : 0 < c₀) (hcsmall : (d+1:ℝ)*c₀ ≤ 1/2) : | |
| let δ := graphCapScale N a p c₀ sig | |
| ∀ᵐ z ∂(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod | |
| (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))), | |
| (∀ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ), | |
| 0 ≤ tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2) ∧ | |
| ∑ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ), | |
| tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2=1 := by | |
| dsimp only | |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae (canonicalSphereMeasureND_ae_norm (d:=d)), | |
| Measure.quasiMeasurePreserving_snd.ae (canonicalSphereMeasureND_ae_norm (d:=d))] with z hz₁ hz₂ | |
| exact sphereCap_pair_partition (graphCapScale_pos hc₀) | |
| ((mul_le_mul_of_nonneg_left (min_le_left _ _) (by positivity)).trans hcsmall) z ⟨hz₁,hz₂⟩ | |
| theorem source_graph_cap_degrees {sig c₀ : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) : | |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (ρ₁ ρ₂ : Measure (EuclideanSpace ℝ (Fin d))) (A₁ A₂ : ℕ → ℝ) | |
| (C : ℝ) (J i₁ i₂ a g₁ g₂ p : ℕ) (X Y : CellPath d) (A B : Finset (CellPath d)) | |
| (b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ), | |
| a ≤ g₁ → a ≤ g₂ → g₁ ≤ p → g₂ ≤ p → | |
| (g₁:ℝ) ≤ (a:ℝ)+2*sig*N → (g₂:ℝ) ≤ (a:ℝ)+2*sig*N → | |
| 0 < J → i₁ ≤ J → i₂ ≤ J → | |
| ImposesParallelTest ρ₁ g₁ p (testWidth N J i₁ p sig) | |
| (((2:ℝ)^N)^(height A₁ g₁ p+C*sig)) b → | |
| ImposesParallelTest ρ₂ g₂ p (testWidth N J i₂ p sig) | |
| (((2:ℝ)^N)^(height A₂ g₂ p+C*sig)) c → | |
| let δ := graphCapScale N a p c₀ sig | |
| let τ := graphTolerance N p sig | |
| let D := ((2:ℝ)^N)^(min (height A₁ a p) (height A₂ a p)+(C+32*(d:ℝ)+1)*sig) | |
| ∀ k ∈ (sphereCapGridND d δ).product (sphereCapGridND d δ), | |
| let Sv := eligibleCells ρ₁ a p X (tensorCapND δ k.1) b | |
| let Sw := eligibleCells ρ₁ a p X (tensorCapND δ k.2) b | |
| let Tv := eligibleCells ρ₂ a p Y (tensorCapND δ k.1) c | |
| let Tw := eligibleCells ρ₂ a p Y (tensorCapND δ k.2) c | |
| let v := sphereCapCenterND δ k.1 | |
| let w := sphereCapCenterND δ k.2 | |
| (∀ row, | |
| (((A.product B).filter (retainedHGraph Sv Tw v w τ row)).card : ℝ) ≤ D ∧ | |
| (((A.product B).filter (retainedHFGraph Sv Sw Tv Tw v w τ row)).card : ℝ) ≤ D) ∧ | |
| (∀ col, | |
| (((A.product B).filter (fun row => retainedHGraph Sv Tw v w τ row col)).card : ℝ) ≤ D ∧ | |
| (((A.product B).filter (fun row => retainedHFGraph Sv Sw Tv Tw v w τ row col)).card : ℝ) ≤ D) := by | |
| obtain ⟨N₀,hN₀⟩ := source_graph_degrees (d:=d) hsig hc₀.le (by positivity : (0:ℝ) ≤ 2*(d+1:ℝ)) | |
| refine ⟨N₀,?_⟩ | |
| intro N hN ρ₁ ρ₂ A₁ A₂ C J i₁ i₂ a g₁ g₂ p X Y A B b c hag₁ hag₂ hg₁ hg₂ hga₁ hga₂ hJ hi₁ hi₂ ht₁ ht₂ δ τ D k hk | |
| have hk' := Finset.mem_product.mp hk | |
| apply hN₀ N hN ρ₁ ρ₂ A₁ A₂ C J i₁ i₂ a g₁ g₂ p X Y A B | |
| (tensorCapND δ k.1) (tensorCapND δ k.2) b c | |
| (sphereCapCenterND δ k.1) (sphereCapCenterND δ k.2) | |
| hag₁ hag₂ hg₁ hg₂ hga₁ hga₂ hJ hi₁ hi₂ (sphereCapCenterND_norm hk'.1) (sphereCapCenterND_norm hk'.2) | |
| ?_ ?_ ht₁ ht₂ | |
| · intro w hw hn | |
| rw [dist_comm] | |
| exact sphereCapCenterND_dist (graphCapScale_pos hc₀) hk'.1 hw (subset_tsupport _ hn) | |
| · intro w hw hn | |
| rw [dist_comm] | |
| exact sphereCapCenterND_dist (graphCapScale_pos hc₀) hk'.2 hw (subset_tsupport _ hn) | |
| lemma graph_profile_ratio {N : ℕ} (_hN : 0 < N) (m : ℝ → ℝ) (a p : ℕ) : | |
| ((2:ℝ)^N)^(-m p)/((2:ℝ)^N)^(-m a) = | |
| ((2:ℝ)^N)^(-((adjustedProfile (d:=d) m N p-adjustedProfile (d:=d) m N a)+ | |
| (d:ℝ)/2*((p:ℝ)-a)/(N:ℝ))) := by | |
| rw [←Real.rpow_sub (by positivity : (0:ℝ) < 2^N)] | |
| congr 1 | |
| unfold adjustedProfile | |
| ring | |
| lemma graph_profile_label_exponent (m₁ m₂ : ℝ → ℝ) (N a p : ℕ) (e : ℝ) : | |
| (m₁ p-m₁ a)+(m₂ p-m₂ a)+4*e = | |
| (adjustedProfile (d:=d) m₁ N p-adjustedProfile (d:=d) m₁ N a)+ | |
| (adjustedProfile (d:=d) m₂ N p-adjustedProfile (d:=d) m₂ N a)+ | |
| (d:ℝ)*((p:ℝ)-a)/(N:ℝ)+4*e := by | |
| unfold adjustedProfile | |
| ring | |
| lemma graphChildren_card_crude (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| {N a p : ℕ} (hp : p ≤ N) (X : CellPath d) : | |
| ((graphChildren ρ a p X).card:ℝ) ≤ ((2:ℝ)^N)^d := by | |
| have hsub : graphChildren ρ a p X ⊆ dyadicPaths p := by | |
| intro P hP | |
| exact (mem_dyadicPaths _ _).mpr (graphChildren_length hP) | |
| have hc := (Finset.card_le_card hsub).trans (dyadicPaths_card_le p) | |
| have hmono : (2^d)^p ≤ (2^d)^N := Nat.pow_le_pow_right (by positivity) hp | |
| have h := (Nat.cast_le (α:=ℝ)).mpr (hc.trans hmono) | |
| simpa only [Nat.cast_pow,Nat.cast_ofNat,pow_right_comm] using h | |
| lemma graphCellLaw_mass_crude (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsProbabilityMeasure ρ] | |
| {R m : ℝ} (hR : 1 ≤ R) (hm : m ≤ d) (P : Set (EuclideanSpace ℝ (Fin d))) : | |
| (graphCellLaw ρ P (R^(-m))).real univ ≤ R^d := by | |
| rw [graphCellLaw_mass _ _ (by positivity)] | |
| have hinv : (R^(-m))⁻¹=R^m := by rw [Real.rpow_neg (by linarith : 0 ≤ R),inv_inv] | |
| rw [hinv] | |
| calc | |
| _ ≤ R^m*1 := mul_le_mul_of_nonneg_left (measureReal_le_one) (by positivity) | |
| _ ≤ R^d := by simpa only [mul_one,Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le hR hm | |
| lemma graph_profile_ratio_crude {R ma mp : ℝ} (hR : 1 ≤ R) (ha : ma ≤ d) (hp : 0 ≤ mp) : | |
| 0 ≤ R^(-mp)/R^(-ma) ∧ R^(-mp)/R^(-ma) ≤ R^d := by | |
| constructor | |
| · positivity | |
| rw [←Real.rpow_sub (by linarith : 0 < R)] | |
| simpa only [Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le hR (show -mp- -ma ≤ (d:ℝ) by linarith) | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |