openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphHFFiniteJetEstimate.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.Ledger | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphCapScaleInvCrude | |
| 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 PlanarFalconer Attempt4FullFar Attempt4Near | |
| open scoped ComplexConjugate RealInnerProductSpace ContDiff | |
| variable {d : ℕ} | |
| theorem graph_HF_finite_jet_estimate {sig c₀ Ggeom : ℝ} | |
| (hsig : 0 < sig) (hc₀ : 0 < c₀) (hcsmall : (d+1:ℝ)*c₀ ≤ 1/2) (hGgeom : 0 ≤ Ggeom) (T : ℝ) : | |
| ∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) (B₀ : ℕ) (s₁ s₂ : ℝ) (m₁ m₂ : ℝ → ℝ), | |
| RegularizedProfileData ρ N B₀ s₁ sig m₁ → RegularizedProfileData κ N B₀ s₂ sig m₂ → | |
| ∀ (a p : ℕ) (v L Ctest K B Pcut : ℝ) (J i₁ i₂ g₁ g₂ : ℕ) | |
| (X Y : CellPath d) (Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ) | |
| (b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ), | |
| a < p → (p:ℝ) ≤ v → v ≤ N → | |
| (p:ℝ)-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 10*sig*N ≤ v-p → | |
| 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 → | |
| X ∈ dyadicPaths a → Y ∈ dyadicPaths a → 0 < ρ (treeCell X) → 0 < κ (treeCell Y) → | |
| ‖treeCenter X-treeCenter Y‖ ≤ Ggeom*(2:ℝ)^(-(a:ℝ)) → | |
| ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → (∀ r, 0 ≤ Ψ r) → | |
| (∫ u : ℝ, ‖(Ψ u:ℂ)‖) ≤ K → (∀ u, ‖iteratedDeriv m (fun u => (Ψ u:ℂ)) u‖ ≤ K) → 0 ≤ K → | |
| let R := (2:ℝ)^N | |
| let A₁ := adjustedProfile (d:=d) m₁ N | |
| let A₂ := adjustedProfile (d:=d) m₂ N | |
| let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig)) | |
| GraphSymbolJet b m S B → GraphSymbolJet c m S B → | |
| 0 ≤ Pcut → ContDiff ℝ ∞ ψ → (∀ w, ‖w‖=1 → ‖ψ w‖ ≤ 1) → | |
| (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ w‖ ≤ Pcut) → | |
| ImposesParallelTest ρ g₁ p (testWidth N J i₁ p sig) (R^(height A₁ g₁ p+Ctest*sig)) b → | |
| ImposesParallelTest κ g₂ p (testWidth N J i₂ p sig) (R^(height A₂ g₂ p+Ctest*sig)) c → | |
| sourceAngularHFND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) a v Ψ ψ | |
| (fun z => sourceAngularTransformND ρ (treeCell X) (R^(-m₁ a)) b z.1 z.2) | |
| (fun z => sourceAngularTransformND κ (treeCell Y) (R^(-m₂ a)) c z.1 z.2) ≤ | |
| R^(-((A₁ p-A₁ a)+(A₂ p-A₂ a))+min (height A₁ a p) (height A₂ a p)+(Ctest+36*(d:ℝ)+5)*sig)* | |
| graphSourceChildMax ρ κ a p v X Y (R^(-m₁ p)) (R^(-m₂ p)) Ψ b c + C*K*Pcut^2*B^4*R^(-T) := by | |
| classical | |
| obtain ⟨m,C,hC,N₁,hdec⟩ := graph_crossed_nonedge_decay (d:=d) hsig hc₀ hGgeom (T+15*(d:ℝ)) | |
| obtain ⟨N₂,hdeg⟩ := source_graph_cap_degrees (d:=d) hsig hc₀ | |
| refine ⟨m,((9*(c₀⁻¹+1))^d)^2*C,by positivity,max 1 (max N₁ N₂),?_⟩ | |
| intro N hN ρ κ B₀ s₁ s₂ m₁ m₂ hρ hκ a p v L Ctest K B Pcut J i₁ i₂ g₁ g₂ X Y Ψ ψ b c | |
| hap hpv hvN hh hL hgap hag₁ hag₂ hg₁ hg₂ hga₁ hga₂ hJ hi₁ hi₂ hX hY hoccX hoccY hcenters | |
| hΨ hsΨ hΨ0 hK hdΨ hK0 R A₁ A₂ S hb hc hPcut hψ hψ1 hdψ ht₁ ht₂ | |
| let := hρ.probability | |
| let := hκ.probability | |
| have hNpos : 0 < N := lt_of_lt_of_le Nat.zero_lt_one (le_trans (le_max_left _ _) hN) | |
| have hNN₁ : N₁ ≤ N := le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hN | |
| have hNN₂ : N₂ ≤ N := le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hN | |
| have hpN : p ≤ N := by exact_mod_cast hpv.trans hvN | |
| have haN := hap.le.trans hpN | |
| have hR : 1 ≤ R := one_le_pow₀ (by norm_num) | |
| have hRp : 0 < R := by positivity | |
| have hc₀1 : c₀ ≤ 1 := by nlinarith [show (0:ℝ) ≤ d by positivity] | |
| let A := graphChildren ρ a p X | |
| let Z := graphChildren κ a p Y | |
| let δ := graphCapScale N a p c₀ sig | |
| let τ := graphTolerance N p sig | |
| let Caps := (sphereCapGridND d δ).product (sphereCapGridND d δ) | |
| let G := fun k : (Fin d → ℤ)×(Fin d → ℤ) => | |
| retainedHFGraph (eligibleCells ρ a p X (tensorCapND δ k.1) b) | |
| (eligibleCells ρ a p X (tensorCapND δ k.2) b) | |
| (eligibleCells κ a p Y (tensorCapND δ k.1) c) | |
| (eligibleCells κ a p Y (tensorCapND δ k.2) c) | |
| (sphereCapCenterND δ k.1) (sphereCapCenterND δ k.2) τ | |
| let η := fun k : (Fin d → ℤ)×(Fin d → ℤ) => fun z : EuclideanSpace ℝ (Fin d)×EuclideanSpace ℝ (Fin d) => | |
| tensorCapND δ k.1 z.1*tensorCapND δ k.2 z.2 | |
| let U := fun (P : CellPath d) (z : ℝ × EuclideanSpace ℝ (Fin d)) => sourceAngularTransformND ρ (treeCell P) (R^(-m₁ p)) b z.1 z.2 | |
| let W := fun (Q : CellPath d) (z : ℝ × EuclideanSpace ℝ (Fin d)) => sourceAngularTransformND κ (treeCell Q) (R^(-m₂ p)) c z.1 z.2 | |
| let lam₁ := R^(-m₁ p)/R^(-m₁ a) | |
| let lam₂ := R^(-m₂ p)/R^(-m₂ a) | |
| let D := R^(min (height A₁ a p) (height A₂ a p)+(Ctest+32*(d:ℝ)+1)*sig) | |
| let ε := C*K*Pcut^2*B^4*(R^d)^4*R^(-(T+15*(d:ℝ)))*(2:ℝ)^v | |
| have hAB : (A.product Z).Nonempty := Finset.nonempty_product.mpr | |
| ⟨graphChildren_nonempty ρ hap.le hX hoccX,graphChildren_nonempty κ hap.le hY hoccY⟩ | |
| have hlabels : (A.card:ℝ)*Z.card ≤ R^((A₁ p-A₁ a)+(A₂ p-A₂ a)+(d:ℝ)*((p:ℝ)-a)/(N:ℝ)+4*(d+1:ℝ)*sig) := by | |
| have h := graphChildren_pair_profile_card ρ κ hap.le hX hY hRp | |
| (fun P hP => hρ.graph_child_lower hpN hP) (fun Q hQ => hκ.graph_child_lower hpN hQ) | |
| (hρ.graph_parent_upper haN hX hoccX) (hκ.graph_parent_upper haN hY hoccY) | |
| rw [graph_profile_label_exponent (d:=d) m₁ m₂ N a p ((d+1:ℝ)*sig)] at h | |
| convert h using 1 | |
| congr 1 | |
| dsimp only [A₁,A₂] | |
| ring | |
| have hdegree := hdeg N hNN₂ ρ κ A₁ A₂ Ctest J i₁ i₂ a g₁ g₂ p X Y A Z b c | |
| hag₁ hag₂ hg₁.le hg₂.le hga₁ hga₂ hJ hi₁ hi₂ ht₁ ht₂ | |
| have hiΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v)) := | |
| (hΨ.continuous.integrable_of_hasCompactSupport | |
| (IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport Ψ) hsΨ)).comp_div (by positivity) | |
| have hbounds₁ : ∀ P ∈ A, ∀ r, ∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)), | |
| ‖U P (r,w)‖ ≤ |(R^(-m₁ p))⁻¹| * ρ.real (treeCell P)*B := by | |
| intro P _ r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hb.norm_le x w hw) | |
| have hbounds₂ : ∀ Q ∈ Z, ∀ r, ∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)), | |
| ‖W Q (r,w)‖ ≤ |(R^(-m₂ p))⁻¹| * κ.real (treeCell Q)*B := by | |
| intro Q _ r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact sourceAngularTransformND_norm_le _ _ _ _ _ _ (fun x => hc.norm_le x w hw) | |
| have hne : ∀ k ∈ Caps, ∀ row ∈ A.product Z, ∀ col ∈ A.product Z, ¬G k row col → | |
| ‖graphCrossedCellTerm ρ κ (R^(-m₁ p)) (R^(-m₂ p)) b c Ψ ψ v δ k.1 k.2 row col‖ ≤ ε := by | |
| intro k hk row hr col hcol hn | |
| have hk' := Finset.mem_product.mp hk | |
| have hd := hdec N hNN₁ a p v L hh hL (sub_nonneg.mpr hpv) ρ κ X Y row col | |
| (R^(-m₁ p)) (R^(-m₂ p)) (R^d) K B Pcut Ψ ψ b c k.1 k.2 hX hY hcenters hr hcol | |
| (by positivity) (by positivity) (by positivity) | |
| (graphCellLaw_mass_crude ρ hR (hρ.profile_le_dim hNpos hpN) _) | |
| (graphCellLaw_mass_crude ρ hR (hρ.profile_le_dim hNpos hpN) _) | |
| (graphCellLaw_mass_crude κ hR (hκ.profile_le_dim hNpos hpN) _) | |
| (graphCellLaw_mass_crude κ hR (hκ.profile_le_dim hNpos hpN) _) | |
| rw [←graphCapScale_eq] at hd | |
| exact hd hk'.1 hk'.2 hΨ hsΨ hK hdΨ hK0 hPcut hψ hdψ hb hc hn | |
| have hmain := graphFiniteHFND_to_child_max d a p v lam₁ lam₂ hNpos | |
| (A₁ p-A₁ a) (A₂ p-A₂ a) | |
| (min (height A₁ a p) (height A₂ a p)+(Ctest+32*(d:ℝ)+1)*sig) (4*(d+1:ℝ)*sig) 1 | |
| (graph_profile_ratio (d:=d) hNpos m₁ a p) (graph_profile_ratio (d:=d) hNpos m₂ a p) | |
| (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) A Z Caps hAB hlabels G η | |
| (graph_pair_partition_ae hc₀ hcsmall) U W | |
| (fun P _ => sourceAngularTransformND_measurable _ _ _ hb.measurable) | |
| (fun Q _ => sourceAngularTransformND_measurable _ _ _ hc.measurable) | |
| (fun P => |(R^(-m₁ p))⁻¹| * ρ.real (treeCell P)*B) | |
| (fun Q => |(R^(-m₂ p))⁻¹| * κ.real (treeCell Q)*B) hbounds₁ hbounds₂ (D:=D) (by positivity) | |
| (by simp only [one_mul]; exact le_rfl) | |
| (fun k hk row _ => ((hdegree k hk).1 row).2) | |
| (fun k hk col _ => ((hdegree k hk).2 col).2) hiΨ hΨ0 hψ.continuous.measurable (by | |
| filter_upwards [canonicalSphereMeasureND_ae_norm (d:=d)] with w hw | |
| exact hψ1 w hw) | |
| (fun k _ row _ col _ => by | |
| simpa only [η,U,W,Complex.ofReal_mul] using | |
| graph_crossed_cell_integrable ρ κ (R^(-m₁ p)) (R^(-m₂ p)) hb hc v hiΨ | |
| ψ hψ.continuous.measurable (by norm_num : (0:ℝ) ≤ 1) hψ1 δ k.1 k.2 row col) | |
| (show 0 ≤ ε by positivity) (by | |
| intro k hk row hr col hc hn | |
| simpa only [graphCrossedCellTerm,η,U,W,Complex.ofReal_mul] using hne k hk row hr col hc hn) | |
| rw [←graph_children_HF_split ρ κ hap.le hX hY hb hc (R^(-m₁ a)) (R^(-m₂ a)) | |
| (by positivity : R^(-m₁ p) ≠ 0) (by positivity : R^(-m₂ p) ≠ 0) v Ψ ψ] at hmain | |
| have hid : graphSourceChildMax ρ κ a p v X Y (R^(-m₁ p)) (R^(-m₂ p)) Ψ b c = | |
| graphChildMaxND d (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))) p v Ψ A Z hAB U W := by | |
| unfold graphSourceChildMax | |
| split_ifs with hx | |
| · rfl | |
| · exact (hx hAB).elim | |
| rw [←hid,one_mul] at hmain | |
| have hexp : -((A₁ p-A₁ a)+(A₂ p-A₂ a))+ | |
| (min (height A₁ a p) (height A₂ a p)+(Ctest+32*(d:ℝ)+1)*sig)+4*(d+1:ℝ)*sig = | |
| -((A₁ p-A₁ a)+(A₂ p-A₂ a))+min (height A₁ a p) (height A₂ a p)+(Ctest+36*(d:ℝ)+5)*sig := by ring | |
| rw [hexp] at hmain | |
| apply hmain.trans | |
| apply add_le_add le_rfl | |
| have herr := graph_source_error_sum (T:=T) ρ κ hρ hκ hNpos haN hpN hvN | |
| hc₀ hc₀1 hsig.le (by positivity : 0 ≤ C*K*Pcut^2*B^4) X Y | |
| convert herr using 1 | |
| ring | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |