| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.Ledger |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.AssemblyInnerEnergyMem |
|
|
| 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 |
| variable {n : ℕ} |
|
|
| lemma uniform_assembly_root_graph (Ctest : ℝ) {sig : ℝ} (hsig : 0<sig) |
| (hsig1 : sig≤1/100) {J : ℕ} (hJ : 0<J) |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (B total : ℕ) (hW : ∀ L∈W,L.length≤B) |
| (Ψ : ℝ → ℝ) (hΨ : ContDiff ℝ ∞ Ψ) (hsupp : tsupport Ψ⊆Ioo (1/4:ℝ) 4) |
| (hΨ0 : ∀ r,0≤Ψ r) (ψ : EuclideanSpace ℝ (Fin (n+1)) → ℂ) |
| (hψ : ContDiff ℝ ∞ ψ) (hψ1 : ∀ w,‖w‖=1 → ‖ψ w‖≤1) : |
| ∃ M : ℕ,∀ Cq : ℝ,0≤Cq → ∃ Cerr : ℝ,0≤Cerr ∧ ∃ N₀ : ℕ, |
| ∀ e₁ e₂ : RecursiveEnvironmentND n,N₀≤e₁.N → e₂.N=e₁.N → |
| e₁.sig=sig → e₂.sig=sig → e₁.C=Ctest → e₂.C=Ctest → |
| e₁.J=J → e₂.J=J → 0<e₁.q → 0<e₂.q → |
| ∀ B₀ s₁ s₂,RegularizedProfileData e₁.ρ e₁.N B₀ s₁ sig e₁.mass → |
| RegularizedProfileData e₂.ρ e₂.N B₀ s₂ sig e₂.mass → |
| e₁.ρ (treeCell [])=1 → e₂.ρ (treeCell [])=1 → |
| ∀ c : ℕ,0<c → (c:ℝ)<(e₁.N:ℝ)/4 → |
| ∀ (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))), |
| Measurable f → Measurable g → (∀ x,‖f x‖≤1) → (∀ x,‖g x‖≤1) → |
| (∀ b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c,L₁ b∈W) → |
| (∀ b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c,L₂ b∈W) → |
| (∑ b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c,(L₁ b).length)≤total → |
| (∑ b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c,(L₂ b).length)≤total → |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) → |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j P.1 w‖≤Cq) → |
| assemblyRootEnergy e₁ e₂ c L₁ L₂ f g Q P Ψ ψ ≤ |
| ((2:ℝ)^e₁.N)^(-e₁.profile c-e₂.profile c+ |
| min (height e₁.profile 0 c) (height e₂.profile 0 c)+(Ctest+36*(n+1:ℝ)+5)*sig)* |
| _root_.OAI.CurrentFalconer.TreeND.graphSourceChildMax e₁.ρ e₂.ρ 0 c e₁.N [] [] |
| (((2:ℝ)^e₁.N)^(-e₁.mass c)) (((2:ℝ)^e₂.N)^(-e₂.mass c)) Ψ |
| (assemblySymbol e₁ c L₁ f Q) (assemblySymbol e₂ c L₂ g P)+ |
| Cerr*((2:ℝ)^e₁.N)^(-2:ℝ) := by |
| let c₀ : ℝ := 1/(4*(n+2:ℝ)) |
| have hc₀ : 0<c₀ := by dsimp [c₀]; positivity |
| have hcsmall : ((n+1:ℕ)+1:ℝ)*c₀≤1/2 := by |
| have he : ((n+1:ℕ)+1:ℝ)*c₀=1/4 := by |
| dsimp [c₀]; push_cast; field_simp; ring |
| rw [he]; norm_num |
| obtain ⟨M,C,hC,N₁,hgraph⟩ := _root_.OAI.CurrentFalconer.TreeND.graph_HF_finite_jet_estimate |
| (d:=n+1) hsig hc₀ hcsmall (le_refl (0:ℝ)) 2 |
| obtain ⟨K,hK0,hK,hdK⟩ := _root_.OAI.CurrentFalconer.graph_fixed_radial_budget hΨ |
| (isCompact_Icc.of_isClosed_subset (isClosed_tsupport Ψ) (hsupp.trans Ioo_subset_Icc_self)) M |
| obtain ⟨Pcut,hPcut,hcut⟩ := _root_.OAI.CurrentFalconer.graph_fixed_angular_budget hψ M |
| refine ⟨M,?_⟩ |
| intro Cq hCq |
| obtain ⟨E,hE,hjets⟩ := uniform_assemblySymbol_jets W M B total hW hsig Cq hCq |
| obtain ⟨N₂,hjets⟩ := Filter.eventually_atTop.mp hjets |
| obtain ⟨N₃,himposes⟩ := uniform_assemblySymbol_imposes (n:=n) hsig hJ |
| refine ⟨C*K*Pcut^2*E^4,by positivity,max N₁ (max N₂ N₃),?_⟩ |
| intro e₁ e₂ hN heN hs₁ hs₂ hC₁ hC₂ hJ₁ hJ₂ hq₁ hq₂ B₀ s₁ s₂ hd₁ hd₂ hroot₁ hroot₂ |
| c hcpos hc L₁ L₂ f g Q P hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ hP |
| let := hd₁.probability |
| let := hd₂.probability |
| have hNc : (c:ℝ)≤e₁.N := by have := Nat.cast_nonneg (α:=ℝ) e₁.N; linarith only [hc,this] |
| have hcN : c≤e₁.N := by exact_mod_cast hNc |
| have hcN₂ : c≤e₂.N := by omega |
| have h2c : 2*c≤e₁.N := by exact_mod_cast (show (2:ℝ)*c≤e₁.N by linarith only [hc,Nat.cast_nonneg (α:=ℝ) e₁.N]) |
| have h2c₂ : 2*c≤e₂.N := by omega |
| have hlist₁ (b) (hb : b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c) : descriptorLength b≤e₁.N/2 := by |
| have hh := assemblyDescriptorList_length e₁.profile hq₁ h2c |
| have hb' := Finset.le_sup (s:=assemblyDescriptorList e₁.profile e₁.N e₁.q c) (f:=descriptorLength) hb |
| omega |
| have hlist₂ (b) (hb : b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c) : descriptorLength b≤e₁.N/2 := by |
| have hh := assemblyDescriptorList_length e₂.profile hq₂ h2c₂ |
| have hb' := Finset.le_sup (s:=assemblyDescriptorList e₂.profile e₂.N e₂.q c) (f:=descriptorLength) hb |
| omega |
| have hj₁ := hjets e₁.N (by omega) e₁ rfl hs₁ hq₁ c (e₁.N/2) hcN hlist₁ L₁ f Q hf hfn hL₁ ht₁ hQ |
| have hj₂ := hjets e₁.N (by omega) e₂ heN hs₂ hq₂ c (e₁.N/2) (by omega) hlist₂ L₂ g P hg hgn hL₂ ht₂ hP |
| have hi₁ := himposes e₁ (by omega) hs₁ hJ₁ inferInstance hq₁ c hcN L₁ f Q |
| have hi₂ := himposes e₂ (by omega) hs₂ hJ₂ inferInstance hq₂ c hcN₂ L₂ g P |
| have hd₁' := (_root_.OAI.CurrentFalconer.TreeND.literal_regularizedProfileData_iff _ _ _ _ _ _).mp hd₁ |
| have hd₂' := (_root_.OAI.CurrentFalconer.TreeND.literal_regularizedProfileData_iff _ _ _ _ _ _).mp hd₂ |
| rw [heN] at hd₂' |
| have hroot₁' : 0<e₁.ρ (_root_.OAI.CurrentFalconer.TreeND.treeCell []) := by |
| have hr : 0<e₁.ρ (treeCell []) := by rw [hroot₁]; exact zero_lt_one |
| simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hr |
| have hroot₂' : 0<e₂.ρ (_root_.OAI.CurrentFalconer.TreeND.treeCell []) := by |
| have hr : 0<e₂.ρ (treeCell []) := by rw [hroot₂]; exact zero_lt_one |
| simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hr |
| have hrange : (c:ℝ)≤(e₁.N:ℝ)-c+2*sig*e₁.N := by |
| nlinarith [Nat.cast_nonneg (α:=ℝ) e₁.N] |
| have hL : ((e₁.N/2:ℕ):ℝ)≤(e₁.N:ℝ)-c+2*sig*e₁.N := by |
| have hh : ((e₁.N/2:ℕ):ℝ)≤(e₁.N:ℝ)/2 := by simpa using (Nat.cast_div_le (α:=ℝ) (m:=e₁.N) (n:=2)) |
| nlinarith [Nat.cast_nonneg (α:=ℝ) e₁.N] |
| have hgap : 10*sig*e₁.N≤(e₁.N:ℝ)-c := by |
| nlinarith [Nat.cast_nonneg (α:=ℝ) e₁.N] |
| have hi₁' := hi₁ |
| have hi₂' := hi₂ |
| simp only [hC₁] at hi₁' |
| simp only [RecursiveEnvironmentND.profile,heN,hC₂] at hi₂' |
| have hh := hgraph e₁.N (by omega) e₁.ρ e₂.ρ B₀ s₁ s₂ e₁.mass e₂.mass hd₁' hd₂' |
| 0 c e₁.N (e₁.N/2:ℕ) Ctest K E Pcut J 1 1 0 0 [] [] Ψ ψ |
| (assemblySymbol e₁ c L₁ f Q) (assemblySymbol e₂ c L₂ g P) |
| hcpos hNc le_rfl (by simpa using hrange) hL hgap (Nat.zero_le _) (Nat.zero_le _) hcpos hcpos |
| (by norm_num only [Nat.cast_zero,zero_add]; positivity) (by norm_num only [Nat.cast_zero,zero_add]; positivity) hJ hJ hJ |
| (by simp [_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths]) |
| (by simp [_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths]) |
| hroot₁' hroot₂' (by simp) hΨ (hsupp.trans Ioo_subset_Icc_self) hΨ0 hK (hdK M le_rfl) hK0 |
| (by simpa only [Real.rpow_natCast] using hj₁) |
| (by simpa only [Real.rpow_natCast] using hj₂) |
| hPcut hψ hψ1 hcut hi₁' hi₂' |
| have hz₁ := assembly_profile_zero e₁ hd₁ |
| have hz₂ := assembly_profile_zero e₂ hd₂ |
| change _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₁.mass e₁.N 0=0 at hz₁ |
| change _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₂.mass e₂.N 0=0 at hz₂ |
| rw [heN] at hz₂ |
| simp only [hz₁,hz₂,sub_zero,Nat.cast_zero,Nat.cast_add,Nat.cast_one] at hh |
| have hpow : -(e₁.profile c+e₂.profile c)+min (height e₁.profile 0 c) (height e₂.profile 0 c)+ |
| (Ctest+36*(n+1:ℝ)+5)*sig = -e₁.profile c-e₂.profile c+min (height e₁.profile 0 c) (height e₂.profile 0 c)+ |
| (Ctest+36*(n+1:ℝ)+5)*sig := by ring |
| have hp₁ : _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₁.mass e₁.N=e₁.profile := rfl |
| have hp₂ : _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e₂.mass e₁.N=e₂.profile := by |
| rw [←heN]; rfl |
| simp only [hp₁,hp₂,hpow] at hh |
| simpa only [assemblyRootEnergy,heN,_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hh |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
| |
| end OAI |
| |
