| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.PrefixShellLeSourceShellSuccessor |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixParallelMem |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixPhasedGlobalBound |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixShell |
| import OAI.MeasureTheory.FalconerCompletion.Shells.ShellEffectiveScale2 |
|
|
| namespace OAI |
|
|
| /-! Dyadic shells and prefix decompositions. -/ |
|
|
| open _root_.OAI.PlanarFalconer |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND |
| open Set MeasureTheory PlanarFalconer GraphLiteralSupport.Attempt4FullFar |
| open scoped ContDiff |
| variable {n : ℕ} |
|
|
| theorem actual_prefix_high_shell (hn : 1 ≤ n) |
| {sig K : ℝ} (hsig : 0 < sig) (hsig₁ : sig ≤ 1) (hK : 32*(n+2:ℝ) ≤ K) |
| (Jsmooth : ℕ) (hJ : 0 < Jsmooth) (ε T : ℝ) (hε : 0 < ε) (hT : 0 < T) |
| (Ψ : ℝ → ℝ) (hΨ : ContDiff ℝ ∞ Ψ) (hsupp : tsupport Ψ ⊆ Ioo (1/4:ℝ) 4) |
| (hΨ0 : ∀ r, 0 ≤ Ψ r) (hΨ1 : ∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) : |
| ∃ Order : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)), |
| N₀ ≤ e.N → e.sig=sig → e.J=Jsmooth → e.i < e.J → |
| ∀ (B : ℕ) (s : ℝ), RegularizedProfileData e.ρ e.N B s sig e.mass → |
| 0 < e.q → e.t ≤ e.N → e.a+100*e.q < e.t → sig*e.N ≤ e.q → |
| 2*e.sourcePrefixDepth ≤ e.a+e.t → |
| e.X.length=e.a → e.Y.length=e.a → 0 < e.ρ (treeCell e.X) → 0 < e.ρ (treeCell e.Y) → |
| farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) → |
| ∀ v : ℕ, v+1 ≤ e.t → (e.t:ℝ)-(v+1:ℕ) < sig*e.N → |
| e.prefixShell v ≤ ((2:ℝ)^e.N)^ε*e.prefixGraphFactor*e.prefixHighChildSup Order (v+1) Ψ + |
| C*((2:ℝ)^e.N)^(-T) := by |
| classical |
| have hKpos : 0 < K := by have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg n; linarith |
| choose D hD hderiv using (fun m : ℕ => uniform_prefixEndpoint_derivatives (n:=n) m) |
| have hsource (Z : Fin (n+1) → ℤ) := |
| Attempt4PrefixShell.source_weighted_shellND (ι:=PolicyDescriptor) (κ:=PolicyDescriptor) |
| (separatedCutoff 0 Z 0) (separatedCutoff_contDiff _ _ _) 3 D sig ε T |
| (2*(n+2:ℝ)*sig) 16 1 (8*K) hsig hsig₁ hD hε (by norm_num) (by positivity) |
| choose order hsource using hsource |
| let offsets := (separatedOffsets_finite (d:=n+1) K).toFinset |
| let Order := offsets.sup order |
| have hξ := (Filter.eventually_all_finite (separatedOffsets_finite (d:=n+1) K)).mpr |
| (fun Z _ => hsource Z) |
| obtain ⟨N₁,hsource⟩ := Filter.eventually_atTop.mp hξ |
| obtain ⟨Cg,hCg,N₂,hgraph⟩ := uniform_actual_prefix_inverse_graph hn hsig hK Jsmooth Order hJ |
| (T+ε) (by linarith) Ψ hΨ hsupp hΨ0 hΨ1 |
| refine ⟨Order,Cg+1,by positivity,max 1 (max N₁ N₂),?_⟩ |
| intro e hN hse hJe hi B s hρ hq htN hwide hqscale hpref hX hY hoccX hoccY hsep v hv hhigh |
| have hN1 : 1 ≤ e.N := (le_max_left _ _).trans hN |
| have hN₁ : N₁ ≤ e.N := ((le_max_left _ _).trans (le_max_right _ _)).trans hN |
| have hN₂ : N₂ ≤ e.N := ((le_max_right _ _).trans (le_max_right _ _)).trans hN |
| let := hρ.probability |
| let := e.weightedDistance_isFinite hK hX hY hsep |
| have haN : e.a ≤ e.N := by omega |
| have hcompact : HasCompactSupport Ψ := |
| IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport Ψ) (hsupp.trans Ioo_subset_Icc_self) |
| have hiΨ : Integrable (fun r => Ψ (r/(2:ℝ)^(v+1:ℕ))) := |
| (hΨ.continuous.integrable_of_hasCompactSupport hcompact).comp_div (by positivity) |
| have hiΨ' : Integrable (fun r => Ψ (r/(2:ℝ)^((v+1:ℕ):ℝ))) := by |
| simpa only [Real.rpow_natCast] using hiΨ |
| have hsup := e.prefixHighChildSup_nonneg Order (v+1:ℕ) Ψ hiΨ' hΨ0 |
| have hfactor : 0 ≤ e.prefixGraphFactor := by unfold PositiveDistanceState.prefixGraphFactor; positivity |
| let BHF := e.prefixGraphFactor*e.prefixHighChildSup Order (v+1:ℕ) Ψ+ |
| Cg*((2:ℝ)^e.N)^(-(T+ε)) |
| have hBHF : 0 ≤ BHF := add_nonneg (mul_nonneg hfactor hsup) (by positivity) |
| have hHF : ∀ j ≤ Order, ∀ z ∈ Attempt4PrefixShell.inverseAngularTermsND j |
| (e.prefixDescriptors.disjSum e.prefixDescriptors), ∀ ξ, e.prefixAngularHF (v+1:ℕ) Ψ z ξ ≤ BHF := by |
| intro j hj z hz ξ |
| apply (hgraph e hN₂ hse hJe hi B s hρ hq htN hwide hqscale hpref hX hY hoccX hoccY hsep |
| (v+1:ℕ) (by exact_mod_cast hv) hhigh j hj z hz ξ).trans |
| exact add_le_add (mul_le_mul_of_nonneg_left |
| (e.prefixGraphChildMax_le_sup Order (v+1:ℕ) Ψ hiΨ' hΨ0 hj hz ξ) hfactor) le_rfl |
| let Z : Fin (n+1) → ℤ := (fun j => (pathIndex e.X j:ℤ))-(fun j => (pathIndex e.Y j:ℤ)) |
| have hZ : Z∈separatedOffsets K := gridOffset_mem hsep |
| have horder : order Z ≤ Order := Finset.le_sup (f:=order) |
| ((separatedOffsets_finite (d:=n+1) K).mem_toFinset.mpr hZ) |
| have hcutEq : separatedCutoff 0 Z 0=e.prefixCutoff := (separatedCutoff_offset e.a _ _).symm |
| have hh := hsource e.N hN₁ Z hZ |
| dsimp only at hh |
| rw [hcutEq,←hse] at hh |
| have hreg := e.prefixHigh_shell_regime hsig htN hwide hqscale hv hhigh |
| have hgeometry : ∀ x∈treeCell e.X, ∀ y∈treeCell e.Y, |
| 1 ≤ dist x y*(2:ℝ)^(e.a:ℝ) ∧ dist x y*(2:ℝ)^(e.a:ℝ) ≤ 8*K := by |
| intro x hx y hy |
| simpa only [Real.rpow_natCast] using e.separated_distance_ratio hK hX hY hsep (z:=(x,y)) hx hy |
| choose G hG hsbound using (fun z : PrefixAssignment n => e.prefixInverse_global_bound z) |
| have hshell := hh e.prefixDescriptors e.prefixDescriptors e.prefixEndpointSmooth e.prefixEndpointSmooth |
| (fun _ => 1) (fun _ => 1) |
| (fun b => sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig) |
| (fun b => sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig) |
| (e.prefixDescriptors_card_max hN1 hq) e.ρ e.ρ (treeCell e.X) (treeCell e.Y) |
| (measurableSet_treeCell _) (measurableSet_treeCell _) ((e.t-e.a)/2) |
| e.levelMass e.levelMass e.a (v+1:ℕ) |
| (by simpa only [hse] using hreg.1) (by simpa only [hse] using hreg.2.1) hreg.2.2 |
| (by simpa only [hse] using e.prefixShell_normalized_mass hρ haN hX hY) |
| (fun b hb => e.prefixDescriptor_scale hq hb) (fun b hb => e.prefixDescriptor_scale hq hb) |
| (fun x _ b _ w hw => e.prefixEndpoint_norm_le x w b hw) |
| (fun y _ b _ w hw => e.prefixEndpoint_norm_le y w b hw) |
| (fun x _ b _ j hj w _ => hderiv _ e x b j hj w) |
| (fun y _ b _ j hj w _ => hderiv _ e y b j hj w) |
| hgeometry (e.ae_prefixCutoff_local (by convert hK using 1; push_cast; ring) hX hY hsep) |
| (Filter.Eventually.of_forall (fun _ => by simp)) |
| (by simpa only [map_one,one_mul,e.prefix_pairMask_radial,Function.comp_def] using |
| Complex.continuous_ofReal.measurable.comp e.measurable_minimalMask) |
| 1 e.prefixCutoff_norm_le (fun _ => G) (fun _ => G) |
| (fun _ _ z _ => e.prefixInverseLeft_measurable z) |
| (fun _ _ z _ => e.prefixInverseRight_measurable z) |
| (fun _ _ z _ x w => (hsbound z x w).1) (fun _ _ z _ y w => (hsbound z y w).2) |
| Ψ hiΨ' hΨ0 hΨ1 BHF hBHF |
| (fun j hj z hz ξ => by |
| simp only [Real.rpow_natCast] |
| exact hHF j (hj.trans horder) z hz ξ) |
| have hmain := (e.prefixShell_le_sourceShell_successor v).trans |
| (by simpa only [Real.rpow_natCast] using hshell.2) |
| apply hmain.trans_eq |
| have hpay : ((2:ℝ)^e.N)^ε*((2:ℝ)^e.N)^(-(T+ε))=((2:ℝ)^e.N)^(-T) := by |
| rw [←Real.rpow_add (by positivity : 0 < (2:ℝ)^e.N)] |
| congr 1 |
| ring |
| dsimp only [BHF] |
| calc |
| _ = ((2:ℝ)^e.N)^ε*e.prefixGraphFactor*e.prefixHighChildSup Order (v+1:ℕ) Ψ+ |
| Cg*(((2:ℝ)^e.N)^ε*((2:ℝ)^e.N)^(-(T+ε)))+((2:ℝ)^e.N)^(-T) := by ring |
| _ = _ := by rw [hpay]; simp only [Nat.cast_add,Nat.cast_one]; ring |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
| |
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND.PositiveDistanceState |
| open Set MeasureTheory |
| variable {d : ℕ} (e : PositiveDistanceState d) |
| lemma prefixHighIndices_spec {j : ℕ} (hj : j ∈ e.prefixHighIndices) : |
| j+1 ≤ e.t ∧ (e.t:ℝ)-(j+1:ℕ) < e.sig*e.N := by |
| have hj' := Finset.mem_filter.mp hj |
| exact ⟨Nat.succ_le_of_lt (Finset.mem_Ico.mp hj'.1).2, lt_of_not_ge hj'.2⟩ |
|
|
| lemma prefixHighIndices_card_bound (htN : e.t ≤ e.N) : e.prefixHighIndices.card ≤ e.N := by |
| calc |
| _ ≤ (Finset.Ico e.a e.t).card := Finset.card_filter_le _ _ |
| _ = e.t-e.a := Nat.card_Ico _ _ |
| _ ≤ e.N := (Nat.sub_le _ _).trans htN |
|
|
| lemma prefixHigh_sum_bound (A C : ℝ) (H : ℕ → ℝ) |
| (h : ∀ j ∈ e.prefixHighIndices, e.prefixShell j ≤ A*H j+C) : |
| (∑ j∈e.prefixHighIndices, e.prefixShell j) ≤ |
| A*(∑ j∈e.prefixHighIndices, H j)+(e.prefixHighIndices.card:ℝ)*C := by |
| calc |
| _ ≤ ∑ j∈e.prefixHighIndices, (A*H j+C) := Finset.sum_le_sum h |
| _ = _ := by rw [Finset.sum_add_distrib,Finset.mul_sum,Finset.sum_const]; simp only [nsmul_eq_mul] |
| |
| lemma prefixHigh_error_absorption (htN : e.t ≤ e.N) (Cs T : ℝ) (hCs : 0 ≤ Cs) : |
| (e.prefixHighIndices.card:ℝ)*(Cs*((2:ℝ)^e.N)^(-(T+1))) ≤ Cs*((2:ℝ)^e.N)^(-T) := by |
| have hNR : (e.N:ℝ) ≤ (2:ℝ)^e.N := by |
| exact_mod_cast Nat.le_of_lt ((Nat.lt_two_pow_self (n:=e.N))) |
| have hpay : (2:ℝ)^e.N*((2:ℝ)^e.N)^(-(T+1))=((2:ℝ)^e.N)^(-T) := by |
| conv_lhs => left; rw [←Real.rpow_one ((2:ℝ)^e.N)] |
| rw [←Real.rpow_add (by positivity : 0 < (2:ℝ)^e.N)] |
| congr 1 |
| ring |
| have hcard : (e.prefixHighIndices.card:ℝ) ≤ e.N := by exact_mod_cast e.prefixHighIndices_card_bound htN |
| calc |
| _ ≤ (2:ℝ)^e.N*(Cs*((2:ℝ)^e.N)^(-(T+1))) := |
| mul_le_mul_of_nonneg_right (hcard.trans hNR) (by positivity) |
| _ = Cs*((2:ℝ)^e.N*((2:ℝ)^e.N)^(-(T+1))) := by ring |
| _ = _ := by rw [hpay] |
| end GraphLiteralSupport.CurrentFalconer.TreeND.PositiveDistanceState |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND |
| lemma prefix_high_accumulation {D H S L X W₁ W₂ Q Cs E : ℝ} (hH : 0 ≤ H) |
| (hd : D ≤ H*(S+L+2*X)) (hx : X ≤ W₁*W₂*Q+Cs*E) : |
| D ≤ H*(S+L+2*W₁*W₂*Q)+(2*H*Cs)*E := by |
| have h := mul_le_mul_of_nonneg_left hx (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) hH) |
| nlinarith only [hd,h] |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND |
| open Set MeasureTheory PlanarFalconer GraphLiteralSupport.Attempt4FullFar |
| open scoped ContDiff |
| variable {n : ℕ} |
|
|
| theorem uniform_actual_prefix_reduction (hn : 1 ≤ n) |
| {sig K : ℝ} (hsig : 0 < sig) (hsig₁ : sig ≤ 1) (hK : 32*(n+2:ℝ) ≤ K) |
| (Jsmooth : ℕ) (hJ : 0 < Jsmooth) (ε T : ℝ) (hε : 0 < ε) (hT : 0 < T) |
| (Ψ : ℝ → ℝ) (hΨ : ContDiff ℝ ∞ Ψ) (hsupp : tsupport Ψ ⊆ Ioo (1/4:ℝ) 4) |
| (hΨ0 : ∀ r, 0 ≤ Ψ r) (hΨ1 : ∀ r ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ r) : |
| ∃ Order : ℕ, ∃ H C : ℝ, 0 < H ∧ 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)), |
| N₀ ≤ e.N → e.sig=sig → e.J=Jsmooth → e.i < e.J → |
| ∀ (B : ℕ) (s : ℝ), RegularizedProfileData e.ρ e.N B s sig e.mass → |
| 0 < e.q → e.t ≤ e.N → e.a+100*e.q < e.t → sig*e.N ≤ e.q → |
| 2*e.sourcePrefixDepth ≤ e.a+e.t → |
| e.X.length=e.a → e.Y.length=e.a → 0 < e.ρ (treeCell e.X) → 0 < e.ρ (treeCell e.Y) → |
| farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) → |
| e.coincidence.toReal ≤ H*(2*((2:ℝ)^e.N)^(4*(n+2:ℝ)*sig)+ |
| 2*(32*binKernelConstant.toReal)* |
| (∑ j∈e.prefixLowIndices, ({e with t:=j+1,i:=e.i+1} : PositiveDistanceState (n+1)).coincidence.toReal)+ |
| 2*((2:ℝ)^e.N)^ε*e.prefixGraphFactor* |
| (∑ j∈e.prefixHighIndices, e.prefixHighChildSup Order (j+1:ℕ) Ψ))+ |
| C*((2:ℝ)^e.N)^(-T) := by |
| classical |
| obtain ⟨H,hH,N₁,hdecomp⟩ := uniform_actual_prefix_decomposition (n:=n) hsig hJ hK |
| obtain ⟨Order,Cs,hCs,N₂,hshell⟩ := actual_prefix_high_shell hn hsig hsig₁ hK |
| Jsmooth hJ ε (T+1) hε (by linarith) Ψ hΨ hsupp hΨ0 hΨ1 |
| refine ⟨Order,H,2*H*Cs,hH,by positivity,max N₁ N₂,?_⟩ |
| intro e hN hse hJe hi B s hρ hq htN hwide hqscale hpref hX hY hoccX hoccY hsep |
| have hN₁ : N₁ ≤ e.N := (le_max_left _ _).trans hN |
| have hN₂ : N₂ ≤ e.N := (le_max_right _ _).trans hN |
| have hat : e.a ≤ e.t := by omega |
| have hd := hdecomp e hN₁ hse hJe hq hat htN hX hY hsep B s sig hρ |
| have hsum := e.prefixHigh_sum_bound (((2:ℝ)^e.N)^ε*e.prefixGraphFactor) |
| (Cs*((2:ℝ)^e.N)^(-(T+1))) (fun j => e.prefixHighChildSup Order (j+1:ℕ) Ψ) (by |
| intro j hj |
| have hj' := e.prefixHighIndices_spec hj |
| simpa only [Nat.cast_add,Nat.cast_one] using hshell e hN₂ hse hJe hi B s hρ hq htN hwide hqscale hpref |
| hX hY hoccX hoccY hsep j hj'.1 (by simpa only [hse] using hj'.2)) |
| have herr := e.prefixHigh_error_absorption htN Cs T hCs |
| have hsum' := hsum.trans (add_le_add_right herr _) |
| exact prefix_high_accumulation hH.le hd hsum' |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
|
|
| end OAI |
|
|
