| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFixedAngularBudget | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory Attempt4FullFar | |
| open scoped ComplexConjugate RealInnerProductSpace ContDiff | |
| variable {d : ℕ} | |
| lemma graph_const_one_jet (j : ℕ) (w : EuclideanSpace ℝ (Fin d)) : | |
| ‖iteratedFDeriv ℝ j (fun _ : EuclideanSpace ℝ (Fin d) => (1:ℂ)) w‖ ≤ 1 := by | |
| rcases j with _ | j | |
| · simp | |
| · simp [iteratedFDeriv_succ_const] | |
| theorem graph_direct_nonedge_decay {sig c₀ : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) (T : ℝ) : | |
| ∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (a p : ℕ) (v L : ℝ), (p:ℝ)-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p → | |
| let R := (2:ℝ)^N | |
| let δ := min c₀ (((2:ℝ)^((p:ℝ)-a))⁻¹*R^(10*sig)) | |
| let τ := (2:ℝ)^(-(p:ℝ))*R^(30*sig) | |
| let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig)) | |
| ∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin d))), ∀ [IsFiniteMeasure ρ] [IsFiniteMeasure κ], | |
| ∀ (X Y : CellPath d) (row col : CellPath d × CellPath d) | |
| (M₁ M₂ H K B : ℝ) (Ψ : ℝ → ℝ) | |
| (b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ) (k₁ k₂ : Fin d → ℤ), | |
| X ∈ dyadicPaths a → Y ∈ dyadicPaths a → | |
| row ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) → | |
| col ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) → | |
| 0 ≤ M₁ → 0 ≤ M₂ → 0 ≤ H → | |
| (graphCellLaw ρ (treeCell row.1) M₁).real univ ≤ H → | |
| (graphCellLaw ρ (treeCell col.1) M₁).real univ ≤ H → | |
| (graphCellLaw κ (treeCell row.2) M₂).real univ ≤ H → | |
| (graphCellLaw κ (treeCell col.2) M₂).real univ ≤ H → | |
| k₁ ∈ sphereCapGridND d δ → k₂ ∈ sphereCapGridND d δ → | |
| ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → | |
| (∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K → | |
| (∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K) → 0 ≤ K → | |
| GraphSymbolJet b m S B → GraphSymbolJet c m S B → | |
| ¬retainedHGraph (eligibleCells ρ a p X (tensorCapND δ k₁) b) | |
| (eligibleCells κ a p Y (tensorCapND δ k₂) c) | |
| (sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) τ row col → | |
| ‖graphDirectCellTerm ρ κ M₁ M₂ b c Ψ v δ k₁ k₂ row col‖ ≤ C*K*B^4*H^4*R^(-T)*(2:ℝ)^v := by | |
| obtain ⟨m,C,hC,N₀,hdec⟩ := graph_cap_four_decay d hsig hc₀ | |
| (by positivity : (0:ℝ) ≤ 2*(d+1:ℝ)) T | |
| refine ⟨m,C,hC,N₀,?_⟩ | |
| intro N hN a p v L hh hL hv R δ τ S ρ κ _ _ X Y row col M₁ M₂ H K B Ψ b c k₁ k₂ | |
| hX hY hr hc hM₁ hM₂ hH hP hP' hQ hQ' hk₁ hk₂ hΨ hsΨ hK hdΨ hK0 hb hcj hne | |
| by_cases hin : row ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product | |
| (eligibleCells κ a p Y (tensorCapND δ k₂) c) ∧ | |
| col ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product | |
| (eligibleCells κ a p Y (tensorCapND δ k₂) c) | |
| · have hn : ¬retainedCenterTests (sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) | |
| (treeCenter row.1) (treeCenter row.2) (treeCenter col.1) (treeCenter col.2) τ := | |
| fun h => hne ⟨hin.1,hin.2,h⟩ | |
| have hn' : ¬(‖(treeCenter row.1-treeCenter col.1)-inner ℝ (treeCenter row.1-treeCenter col.1) | |
| (sphereCapCenterND δ k₁) • sphereCapCenterND δ k₁‖ ≤ τ ∧ | |
| ‖(treeCenter row.2-treeCenter col.2)-inner ℝ (treeCenter row.2-treeCenter col.2) | |
| (sphereCapCenterND δ k₂) • sphereCapCenterND δ k₂‖ ≤ τ ∧ | |
| |inner ℝ (sphereCapCenterND δ k₁) (treeCenter row.1-treeCenter col.1)+ | |
| inner ℝ (sphereCapCenterND δ k₂) (treeCenter row.2-treeCenter col.2)| ≤ τ) := by | |
| simpa only [retainedCenterTests,graphPerpND,real_inner_comm] using hn | |
| have hr' := Finset.mem_product.mp hr | |
| have hc' := Finset.mem_product.mp hc | |
| have hgeo : ∀ᵐ z ∂((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw ρ (treeCell col.1) M₁)).prod | |
| ((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw κ (treeCell col.2) M₂)), | |
| ‖z.1.1-z.1.2‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(a:ℝ)) ∧ | |
| ‖z.2.1-z.2.2‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(a:ℝ)) ∧ | |
| ‖(z.1.1-z.1.2)-(treeCenter row.1-treeCenter col.1)‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(p:ℝ)) ∧ | |
| ‖(z.2.1-z.2.2)-(treeCenter row.2-treeCenter col.2)‖ ≤ 2*(d+1:ℝ)*(2:ℝ)^(-(p:ℝ)) := by | |
| filter_upwards [graphCellLaw_four_ae ρ ρ κ κ (treeCell row.1) (treeCell col.1) (treeCell row.2) (treeCell col.2) | |
| M₁ M₁ M₂ M₂ (measurableSet_treeCell row.1) | |
| (measurableSet_treeCell col.1) (measurableSet_treeCell row.2) (measurableSet_treeCell col.2)] with z hz | |
| refine ⟨?_,?_,graphChildren_center_error hr'.1 hc'.1 hz.1 hz.2.1, | |
| graphChildren_center_error hr'.2 hc'.2 hz.2.2.1 hz.2.2.2⟩ | |
| · exact (graphChildren_direct_size hX hr'.1 hc'.1 hz.1 hz.2.1).trans (by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(a:ℝ))]) | |
| · exact (graphChildren_direct_size hY hr'.2 hc'.2 hz.2.2.1 hz.2.2.2).trans (by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(a:ℝ))]) | |
| have hsΨ' : tsupport (fun u => (Ψ u:ℂ)) ⊆ Icc (1/4:ℝ) 4 := by | |
| have he : tsupport (fun u => (Ψ u:ℂ)) = tsupport Ψ := by | |
| change closure (Function.support _) = closure (Function.support _) | |
| congr 1 | |
| ext u | |
| simp only [Function.mem_support,ne_eq,Complex.ofReal_eq_zero] | |
| rwa [he] | |
| have h := hdec N hN a p v L hh hL hv | |
| (graphCellLaw ρ (treeCell row.1) M₁) (graphCellLaw ρ (treeCell col.1) M₁) | |
| (graphCellLaw κ (treeCell row.2) M₂) (graphCellLaw κ (treeCell col.2) M₂) | |
| (fun u => (Ψ u:ℂ)) K B 1 k₁ k₂ (treeCenter row.1-treeCenter col.1) | |
| (treeCenter row.2-treeCenter col.2) (fun _ => 1) (fun _ => 1) b b c c | |
| hk₁ hk₂ (Complex.ofRealCLM.contDiff.comp hΨ) hsΨ' hK hdΨ hK0 (by norm_num) | |
| contDiff_const contDiff_const (fun j _ w _ => graph_const_one_jet j w) | |
| (fun j _ w _ => graph_const_one_jet j w) hb hb hcj hcj hgeo hn' | |
| simp only [graphCellLaw_transform _ _ hM₁,graphCellLaw_transform _ _ hM₂] at h | |
| rw [←graphDirectCellTerm_eq] at h | |
| simp only [one_pow,mul_one] at h | |
| have hmass : (((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw ρ (treeCell col.1) M₁)).prod | |
| ((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw κ (treeCell col.2) M₂))).real univ ≤ H^4 := by | |
| simp only [Measure.real,←Set.univ_prod_univ,Measure.prod_prod,ENNReal.toReal_mul] | |
| calc | |
| _ ≤ (H*H)*(H*H) := mul_le_mul (mul_le_mul hP hP' measureReal_nonneg hH) | |
| (mul_le_mul hQ hQ' measureReal_nonneg hH) (by positivity) (by positivity) | |
| _ = H^4 := by ring | |
| calc | |
| _ ≤ _ := h | |
| _ ≤ C*K*B^4*R^(-T)*(2:ℝ)^v*H^4 := mul_le_mul_of_nonneg_left hmass (by positivity) | |
| _ = _ := by ring | |
| · rw [graph_direct_ineligible_zero ρ κ hr hc M₁ M₂ hb hcj Ψ v δ k₁ k₂ hin,norm_zero] | |
| positivity | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory Attempt4FullFar | |
| open scoped ComplexConjugate RealInnerProductSpace ContDiff | |
| variable {d : ℕ} | |
| theorem graph_crossed_nonedge_decay {sig c₀ G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) (hG : 0 ≤ G) (T : ℝ) : | |
| ∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (a p : ℕ) (v L : ℝ), (p:ℝ)-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p → | |
| let R := (2:ℝ)^N | |
| let δ := min c₀ (((2:ℝ)^((p:ℝ)-a))⁻¹*R^(10*sig)) | |
| let τ := (2:ℝ)^(-(p:ℝ))*R^(30*sig) | |
| let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig)) | |
| ∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin d))), ∀ [IsFiniteMeasure ρ] [IsFiniteMeasure κ], | |
| ∀ (X Y : CellPath d) (row col : CellPath d × CellPath d) | |
| (M₁ M₂ H K B P : ℝ) (Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ) | |
| (b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ) (k₁ k₂ : Fin d → ℤ), | |
| X ∈ dyadicPaths a → Y ∈ dyadicPaths a → | |
| ‖treeCenter X-treeCenter Y‖ ≤ G*(2:ℝ)^(-(a:ℝ)) → | |
| row ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) → | |
| col ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y) → | |
| 0 ≤ M₁ → 0 ≤ M₂ → 0 ≤ H → | |
| (graphCellLaw ρ (treeCell row.1) M₁).real univ ≤ H → | |
| (graphCellLaw ρ (treeCell col.1) M₁).real univ ≤ H → | |
| (graphCellLaw κ (treeCell row.2) M₂).real univ ≤ H → | |
| (graphCellLaw κ (treeCell col.2) M₂).real univ ≤ H → | |
| k₁ ∈ sphereCapGridND d δ → k₂ ∈ sphereCapGridND d δ → | |
| ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → | |
| (∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K → | |
| (∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K) → 0 ≤ K → | |
| 0 ≤ P → ContDiff ℝ ∞ ψ → | |
| (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ w‖ ≤ P) → | |
| GraphSymbolJet b m S B → GraphSymbolJet c m S B → | |
| ¬retainedHFGraph (eligibleCells ρ a p X (tensorCapND δ k₁) b) | |
| (eligibleCells ρ a p X (tensorCapND δ k₂) b) | |
| (eligibleCells κ a p Y (tensorCapND δ k₁) c) | |
| (eligibleCells κ a p Y (tensorCapND δ k₂) c) | |
| (sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) τ row col → | |
| ‖graphCrossedCellTerm ρ κ M₁ M₂ b c Ψ ψ v δ k₁ k₂ row col‖ ≤ C*K*P^2*B^4*H^4*R^(-T)*(2:ℝ)^v := by | |
| obtain ⟨m,C,hC,N₀,hdec⟩ := graph_cap_four_decay d hsig hc₀ | |
| (by positivity : (0:ℝ) ≤ 2*(d+1:ℝ)+G) T | |
| refine ⟨m,C,hC,N₀,?_⟩ | |
| intro N hN a p v L hh hL hv R δ τ S ρ κ _ _ X Y row col M₁ M₂ H K B P Ψ ψ b c k₁ k₂ | |
| hX hY hcenters hr hc hM₁ hM₂ hH hP hP' hQ hQ' hk₁ hk₂ hΨ hsΨ hK hdΨ hK0 hP0 hψ hdψ hb hcj hne | |
| by_cases hin : row ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product | |
| (eligibleCells κ a p Y (tensorCapND δ k₂) c) ∧ | |
| col ∈ (eligibleCells ρ a p X (tensorCapND δ k₂) b).product | |
| (eligibleCells κ a p Y (tensorCapND δ k₁) c) | |
| · have hn : ¬retainedCenterTests (sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) | |
| (treeCenter row.1) (treeCenter row.2) (treeCenter col.2) (treeCenter col.1) τ := | |
| fun h => hne ⟨hin.1,hin.2,h⟩ | |
| have hn' : ¬(‖(treeCenter row.1-treeCenter col.2)-inner ℝ (treeCenter row.1-treeCenter col.2) | |
| (sphereCapCenterND δ k₁) • sphereCapCenterND δ k₁‖ ≤ τ ∧ | |
| ‖(treeCenter row.2-treeCenter col.1)-inner ℝ (treeCenter row.2-treeCenter col.1) | |
| (sphereCapCenterND δ k₂) • sphereCapCenterND δ k₂‖ ≤ τ ∧ | |
| |inner ℝ (sphereCapCenterND δ k₁) (treeCenter row.1-treeCenter col.2)+ | |
| inner ℝ (sphereCapCenterND δ k₂) (treeCenter row.2-treeCenter col.1)| ≤ τ) := by | |
| simpa only [retainedCenterTests,graphPerpND,real_inner_comm] using hn | |
| have hr' := Finset.mem_product.mp hr | |
| have hc' := Finset.mem_product.mp hc | |
| have hgeo : ∀ᵐ z ∂((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw κ (treeCell col.2) M₂)).prod | |
| ((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw ρ (treeCell col.1) M₁)), | |
| ‖z.1.1-z.1.2‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(a:ℝ)) ∧ | |
| ‖z.2.1-z.2.2‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(a:ℝ)) ∧ | |
| ‖(z.1.1-z.1.2)-(treeCenter row.1-treeCenter col.2)‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(p:ℝ)) ∧ | |
| ‖(z.2.1-z.2.2)-(treeCenter row.2-treeCenter col.1)‖ ≤ (2*(d+1:ℝ)+G)*(2:ℝ)^(-(p:ℝ)) := by | |
| filter_upwards [graphCellLaw_four_ae ρ κ κ ρ (treeCell row.1) (treeCell col.2) (treeCell row.2) (treeCell col.1) | |
| M₁ M₂ M₂ M₁ (measurableSet_treeCell row.1) | |
| (measurableSet_treeCell col.2) (measurableSet_treeCell row.2) (measurableSet_treeCell col.1)] with z hz | |
| have hrev : ‖treeCenter Y-treeCenter X‖ ≤ G*(2:ℝ)^(-(a:ℝ)) := by rwa [norm_sub_rev] | |
| refine ⟨graphChildren_cross_size hX hY hr'.1 hc'.2 hcenters hz.1 hz.2.1, | |
| graphChildren_cross_size hY hX hr'.2 hc'.1 hrev hz.2.2.1 hz.2.2.2,?_,?_⟩ | |
| · exact (graphChildren_center_error hr'.1 hc'.2 hz.1 hz.2.1).trans | |
| (by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(p:ℝ))]) | |
| · exact (graphChildren_center_error hr'.2 hc'.1 hz.2.2.1 hz.2.2.2).trans | |
| (by nlinarith [Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) (-(p:ℝ))]) | |
| have hsΨ' : tsupport (fun u => (Ψ u:ℂ)) ⊆ Icc (1/4:ℝ) 4 := by | |
| have he : tsupport (fun u => (Ψ u:ℂ)) = tsupport Ψ := by | |
| change closure (Function.support _) = closure (Function.support _) | |
| congr 1 | |
| ext u | |
| simp only [Function.mem_support,ne_eq,Complex.ofReal_eq_zero] | |
| rwa [he] | |
| have h := hdec N hN a p v L hh hL hv | |
| (graphCellLaw ρ (treeCell row.1) M₁) (graphCellLaw κ (treeCell col.2) M₂) | |
| (graphCellLaw κ (treeCell row.2) M₂) (graphCellLaw ρ (treeCell col.1) M₁) | |
| (fun u => (Ψ u:ℂ)) K B P k₁ k₂ (treeCenter row.1-treeCenter col.2) | |
| (treeCenter row.2-treeCenter col.1) ψ (fun w => conj (ψ w)) b c c b | |
| hk₁ hk₂ (Complex.ofRealCLM.contDiff.comp hΨ) hsΨ' hK hdΨ hK0 hP0 | |
| hψ (Complex.conjCLE.contDiff.comp hψ) hdψ (by | |
| intro j hj w hw | |
| change ‖iteratedFDeriv ℝ j (Complex.conjLIE ∘ ψ) w‖ ≤ P | |
| rw [Complex.conjLIE.norm_iteratedFDeriv_comp_left ψ w j] | |
| exact hdψ j hj w hw) hb hcj hcj hb hgeo hn' | |
| simp only [graphCellLaw_transform _ _ hM₁,graphCellLaw_transform _ _ hM₂] at h | |
| rw [←graphCrossedCellTerm_eq] at h | |
| have hmass : (((graphCellLaw ρ (treeCell row.1) M₁).prod (graphCellLaw κ (treeCell col.2) M₂)).prod | |
| ((graphCellLaw κ (treeCell row.2) M₂).prod (graphCellLaw ρ (treeCell col.1) M₁))).real univ ≤ H^4 := by | |
| simp only [Measure.real,←Set.univ_prod_univ,Measure.prod_prod,ENNReal.toReal_mul] | |
| calc | |
| _ ≤ (H*H)*(H*H) := mul_le_mul (mul_le_mul hP hQ' measureReal_nonneg hH) | |
| (mul_le_mul hQ hP' measureReal_nonneg hH) (by positivity) (by positivity) | |
| _ = H^4 := by ring | |
| calc | |
| _ ≤ _ := h | |
| _ ≤ C*K*P^2*B^4*R^(-T)*(2:ℝ)^v*H^4 := mul_le_mul_of_nonneg_left hmass (by positivity) | |
| _ = _ := by ring | |
| · rw [graph_crossed_ineligible_zero ρ κ hr hc M₁ M₂ hb hcj Ψ ψ v δ k₁ k₂ hin,norm_zero] | |
| positivity | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |