| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.InteriorPolicy |
| import OAI.MeasureTheory.Falconer.Recursion.Ledger |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphHFFiniteJetEstimate |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.LiteralDyadicPaths |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.PrefixDepth |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixCutoff |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixSpatialLeft |
|
|
| 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 Filter PlanarFalconer |
| open scoped ContDiff Topology |
| variable {n : ℕ} |
|
|
| lemma source_prefix_parallel_mem (A : ℕ → ℝ) {N q a t : ℕ} |
| (hq : 0 < q) (ht : t ≤ N) (hw : a+100*q < t) |
| (hp : 2*policyDepth A N q a t ≤ a+t) : |
| PolicyDescriptor.parallel a (policyDepth A N q a t) ∈ policyDescriptorList A N q a t := by |
| classical |
| have hi := policyDepth_interior A hq (policyBlocks_nonempty hq ht hw) |
| apply Finset.mem_biUnion.mpr |
| refine ⟨(a,t),?_,?_⟩ |
| · exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr |
| ⟨Finset.mem_range.mpr (by omega),Finset.mem_range.mpr (by omega)⟩, |
| le_rfl,le_rfl,hw⟩ |
| · apply Finset.mem_union_left |
| apply Finset.mem_image.mpr |
| refine ⟨a,Finset.mem_Ico.mpr ⟨?_,?_⟩,rfl⟩ <;> dsimp only at * <;> omega |
| |
| lemma prefix_high_graph_ranges {a p t q N : ℕ} {sig v : ℝ} |
| (hsig : 0 < sig) (hq : 0 < q) (hN : 0 < N) |
| (hi : a+q ≤ p ∧ p ≤ t-q) (hw : a+100*q < t) |
| (hqt : sig*N ≤ q) (hp : 2*p ≤ a+t) (ht : t ≤ N) |
| (hv : v ≤ t) (hhigh : (t:ℝ)-v < sig*N) : |
| a < p ∧ (p:ℝ) ≤ v ∧ v ≤ N ∧ |
| (p:ℝ)-a ≤ v-p+2*sig*N ∧ |
| (((t-a)/2:ℕ):ℝ) ≤ v-p+2*sig*N ∧ 10*sig*N ≤ v-p := by |
| have ha : a ≤ t := by omega |
| have hp' : (2:ℝ)*p ≤ a+t := by exact_mod_cast hp |
| have hw' : (a:ℝ)+100*q < t := by exact_mod_cast hw |
| have hN' : (0:ℝ) < N := by exact_mod_cast hN |
| have hsn : 0 < sig*N := mul_pos hsig hN' |
| have hfloor : (2:ℝ)*(((t-a)/2:ℕ):ℝ) ≤ (t:ℝ)-a := by |
| have hh : (2:ℝ)*(((t-a)/2:ℕ):ℝ) ≤ ((t-a:ℕ):ℝ) := by |
| exact_mod_cast Nat.mul_div_le (t-a) 2 |
| rwa [Nat.cast_sub ha] at hh |
| refine ⟨by omega,by linarith, hv.trans (by exact_mod_cast ht),?_,?_,?_⟩ <;> linarith |
| |
| namespace PositiveDistanceState |
| variable (e : PositiveDistanceState (n+1)) |
|
|
| lemma prefixInverseLeft_contDiff (z : PrefixAssignment n) (x : EuclideanSpace ℝ (Fin (n+1))) : |
| ContDiff ℝ ∞ (e.prefixInverseLeft z x) := by |
| have h : ContDiff ℝ ∞ (actualAngularProduct e.prefixDescriptors id (prefixLeftWords z) |
| e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := |
| (actualAngularProduct e.prefixDescriptors id (prefixLeftWords z) |
| e.ρ e.mass e.N e.sig e.C e.J e.i x).2 |
| convert h using 1 |
| funext w |
| exact e.prefixInverseLeft_eq z x w |
|
|
| lemma prefixInverseRight_contDiff (z : PrefixAssignment n) (x : EuclideanSpace ℝ (Fin (n+1))) : |
| ContDiff ℝ ∞ (e.prefixInverseRight z x) := by |
| have h : ContDiff ℝ ∞ (actualAngularProduct e.prefixDescriptors id (prefixRightWords z) |
| e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := |
| (actualAngularProduct e.prefixDescriptors id (prefixRightWords z) |
| e.ρ e.mass e.N e.sig e.C e.J e.i x).2 |
| convert h using 1 |
| funext w |
| exact e.prefixInverseRight_eq z x w |
|
|
| lemma prefixPhasedLeft_measurable (z : PrefixAssignment n) (ξ : EuclideanSpace ℝ (Fin (n+1))) : |
| Measurable (Function.uncurry (e.prefixPhasedLeft z ξ)) := by |
| exact ((Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase_continuous (n+1) |
| ((2:ℝ)^e.a)⁻¹).measurable.comp (measurable_const.prodMk measurable_fst)).mul |
| (e.prefixInverseLeft_measurable z) |
|
|
| lemma prefixPhasedRight_measurable (z : PrefixAssignment n) (ξ : EuclideanSpace ℝ (Fin (n+1))) : |
| Measurable (Function.uncurry (e.prefixPhasedRight z ξ)) := by |
| exact ((Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase_continuous (n+1) |
| ((2:ℝ)^e.a)⁻¹).measurable.comp (measurable_const.prodMk measurable_fst)).mul |
| (e.prefixInverseRight_measurable z) |
|
|
| lemma prefixPhased_jets {m : ℕ} {S B : ℝ} (hB : 0 ≤ B) |
| (z : PrefixAssignment n) (ξ : EuclideanSpace ℝ (Fin (n+1))) |
| (hleft : ∀ x r, r ≤ m → ∀ w, ‖w‖=1 → |
| ‖iteratedFDeriv ℝ r (e.prefixInverseLeft z x) w‖ ≤ B*S^r) |
| (hright : ∀ x r, r ≤ m → ∀ w, ‖w‖=1 → |
| ‖iteratedFDeriv ℝ r (e.prefixInverseRight z x) w‖ ≤ B*S^r) : |
| _root_.OAI.CurrentFalconer.GraphSymbolJet (e.prefixPhasedLeft z ξ) m S B ∧ |
| _root_.OAI.CurrentFalconer.GraphSymbolJet (e.prefixPhasedRight z ξ) m S B := by |
| constructor |
| · refine ⟨e.prefixPhasedLeft_measurable z ξ,fun x => contDiff_const.mul |
| (e.prefixInverseLeft_contDiff z x),hB,?_⟩ |
| intro x r hr w hw |
| exact (symbol_const_mul_derivative_boundND ⟨e.prefixInverseLeft z x,e.prefixInverseLeft_contDiff z x⟩ |
| _ (le_of_eq (Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase_norm _ _ _ _)) r w).trans |
| (hleft x r hr w hw) |
| · refine ⟨e.prefixPhasedRight_measurable z ξ,fun x => contDiff_const.mul |
| (e.prefixInverseRight_contDiff z x),hB,?_⟩ |
| intro x r hr w hw |
| exact (symbol_const_mul_derivative_boundND ⟨e.prefixInverseRight z x,e.prefixInverseRight_contDiff z x⟩ |
| _ (le_of_eq (Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase_norm _ _ _ _)) r w).trans |
| (hright x r hr w hw) |
|
|
| lemma prefixAngularHF_graph_eq (v : ℝ) (Ψ : ℝ → ℝ) (z : PrefixAssignment n) |
| (ξ : EuclideanSpace ℝ (Fin (n+1))) : |
| e.prefixAngularHF v Ψ z ξ = |
| _root_.OAI.Attempt4Near.sourceAngularHFND (n+1) |
| (_root_.OAI.Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) |
| e.a v Ψ e.prefixCutoff |
| (fun rw => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ |
| (_root_.OAI.CurrentFalconer.TreeND.treeCell e.X) (((2:ℝ)^e.N)^(-e.mass e.a)) |
| (e.prefixPhasedLeft z ξ) rw.1 rw.2) |
| (fun rw => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ |
| (_root_.OAI.CurrentFalconer.TreeND.treeCell e.Y) (((2:ℝ)^e.N)^(-e.mass e.a)) |
| (e.prefixPhasedRight z ξ) rw.1 rw.2) := by |
| simp only [prefixAngularHF,prefixSpatialLeft,prefixSpatialRight, |
| Attempt4PrefixShell.spatialEndpointTransformND,Attempt4PrefixShell.sourceAngularHFND, |
| _root_.OAI.Attempt4Near.sourceAngularHFND,Attempt4PrefixShell.sourceAngularTransformND, |
| _root_.OAI.CurrentFalconer.sourceAngularTransformND,levelMass, |
| _root_.OAI.CurrentFalconer.TreeND.literal_treeCell] |
| rfl |
|
|
| lemma prefix_graph_centers {K : ℝ} (hX : e.X.length=e.a) (hY : e.Y.length=e.a) |
| (hs : Attempt4FullFar.farSeparatedGridND e.a K |
| (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) : |
| ‖_root_.OAI.CurrentFalconer.TreeND.treeCenter e.X-_root_.OAI.CurrentFalconer.TreeND.treeCenter e.Y‖ ≤ |
| (4*K)*(2:ℝ)^(-(e.a:ℝ)) := by |
| rw [←_root_.OAI.CurrentFalconer.TreeND.literal_treeCenter, |
| ←_root_.OAI.CurrentFalconer.TreeND.literal_treeCenter] |
| change ‖ambientGridCenter e.X.length _-ambientGridCenter e.Y.length _‖ ≤ _ |
| rw [hX,hY,←dist_eq_norm,Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),Real.rpow_natCast] |
| simpa only [div_eq_mul_inv,ambientGridCenter,Attempt4FullFar.ambientGridCenter] using hs.2 |
| end PositiveDistanceState |
|
|
| lemma uniform_prefixPhased_imposes {sig : ℝ} (hsig : 0 < sig) {J : ℕ} (hJ : 0 < J) : |
| ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)), N₀ ≤ e.N → e.sig=sig → e.J=J → |
| IsFiniteMeasure e.ρ → 0 < e.q → e.t ≤ e.N → e.a+100*e.q < e.t → |
| 2*e.sourcePrefixDepth ≤ e.a+e.t → ∀ (z : PrefixAssignment n) |
| (ξ : EuclideanSpace ℝ (Fin (n+1))), |
| let bound := ((2:ℝ)^e.N)^(height (adjustedProfile (d:=n+1) e.mass e.N) |
| e.a e.sourcePrefixDepth+e.C*sig) |
| let width := _root_.OAI.CurrentFalconer.TreeND.testWidth e.N J (e.i+1) e.sourcePrefixDepth sig |
| _root_.OAI.CurrentFalconer.TreeND.ImposesParallelTest e.ρ e.a e.sourcePrefixDepth width bound |
| (e.prefixPhasedLeft z ξ) ∧ |
| _root_.OAI.CurrentFalconer.TreeND.ImposesParallelTest e.ρ e.a e.sourcePrefixDepth width bound |
| (e.prefixPhasedRight z ξ) := by |
| obtain ⟨N₀,hsupport⟩ := uniform_prefixInverse_support (n:=n) hsig hJ |
| refine ⟨N₀,?_⟩ |
| intro e hN he hJe hf hq ht hw hp z ξ |
| have hmem : PolicyDescriptor.parallel e.a e.sourcePrefixDepth ∈ e.prefixDescriptors := |
| source_prefix_parallel_mem _ hq ht hw hp |
| have hh (x w : EuclideanSpace ℝ (Fin (n+1))) (hw : ‖w‖=1) := |
| hsupport e hN he hJe hf hq ht z 0 x w hw |
| have htest (x w : EuclideanSpace ℝ (Fin (n+1))) |
| (h : (x,w) ∈ endpointGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) |
| (.parallel e.a e.sourcePrefixDepth)) |
| (P : _root_.OAI.CurrentFalconer.TreeND.CellPath (n+1)) |
| (hP : P ∈ _root_.OAI.CurrentFalconer.TreeND.dyadicPaths e.sourcePrefixDepth) |
| (hx : x ∈ _root_.OAI.CurrentFalconer.TreeND.treeCell P) : |
| (_root_.OAI.CurrentFalconer.TreeND.fullParallelCount e.ρ e.a e.sourcePrefixDepth P |
| (_root_.OAI.CurrentFalconer.TreeND.testWidth e.N J (e.i+1) e.sourcePrefixDepth sig) w:ℝ) ≤ |
| ((2:ℝ)^e.N)^(height (adjustedProfile (d:=n+1) e.mass e.N) |
| e.a e.sourcePrefixDepth+e.C*sig) := by |
| have hh := h P (by simpa only [descriptorAnchor,_root_.OAI.CurrentFalconer.TreeND.literal_dyadicPaths] using hP) |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hx) |
| simpa only [descriptorGood,he,hJe,testWidth,_root_.OAI.CurrentFalconer.TreeND.testWidth, |
| _root_.OAI.CurrentFalconer.TreeND.literal_fullParallelCount] using hh.2 |
| constructor |
| · intro P hP x hx w hw hn |
| apply htest x w _ P hP hx |
| apply (hh x w hw).1 _ _ hmem |
| apply subset_closure |
| rw [Function.mem_support,←norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff] |
| exact (mul_ne_zero_iff.mp hn).2 |
| · intro P hP x hx w hw hn |
| apply htest x w _ P hP hx |
| apply (hh x w hw).2 _ _ hmem |
| apply subset_closure |
| rw [Function.mem_support,←norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff] |
| exact (mul_ne_zero_iff.mp hn).2 |
| 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_inverse_graph (hn : 1 ≤ n) |
| {sig K : ℝ} (hsig : 0 < sig) (hK : 32*(n+2:ℝ) ≤ K) |
| (Jsmooth Order : ℕ) (hJ : 0 < Jsmooth) (T : ℝ) (_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) : |
| ∃ 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 : ℝ, v ≤ e.t → (e.t:ℝ)-v < sig*e.N → |
| ∀ j ≤ Order, ∀ z ∈ GraphLiteralSupport.Attempt4PrefixShell.inverseAngularTermsND j |
| (e.prefixDescriptors.disjSum e.prefixDescriptors), |
| ∀ ξ : EuclideanSpace ℝ (Fin (n+1)), |
| e.prefixAngularHF v Ψ z ξ ≤ e.prefixGraphFactor * e.prefixGraphChildMax v Ψ z ξ + |
| C*((2:ℝ)^e.N)^(-T) := by |
| classical |
| 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 |
| have hK0 : 0 ≤ 4*K := by |
| have hn0 : (0:ℝ) ≤ n := by positivity |
| nlinarith |
| obtain ⟨m,C,hC,N₁,hgraph⟩ := _root_.OAI.CurrentFalconer.TreeND.graph_HF_finite_jet_estimate |
| (d:=n+1) hsig hc₀ hcsmall hK0 T |
| obtain ⟨BΨ,hBΨ,hintΨ,hderivΨ⟩ := _root_.OAI.CurrentFalconer.graph_fixed_radial_budget hΨ |
| (IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport Ψ) |
| (hsupp.trans Ioo_subset_Icc_self)) m |
| obtain ⟨P,hP,hcut⟩ := uniform_prefixCutoff_derivatives (d:=n+1) K |
| let Pcut : ℝ := ∑ j ∈ Finset.range (m+1), P j |
| have hPcut : 0 ≤ Pcut := Finset.sum_nonneg (fun j _ => hP j) |
| have hPbound (j : ℕ) (hj : j ≤ m) : P j ≤ Pcut := |
| Finset.single_le_sum (fun k _ => hP k) (Finset.mem_range.mpr (Nat.lt_succ_of_le hj)) |
| obtain ⟨D,hD,hder⟩ := uniform_prefixInverse_derivatives (n:=n) m Order hsig |
| obtain ⟨N₂,hder⟩ := Filter.eventually_atTop.mp hder |
| obtain ⟨N₃,himpose⟩ := uniform_prefixPhased_imposes (n:=n) hsig hJ |
| refine ⟨C*BΨ*Pcut^2*(D^(2*Order))^4,by positivity, |
| max 1 (max N₁ (max N₂ N₃)),?_⟩ |
| intro e he hse hJe hi B s hρ hq ht hwide hqscale hprefix hX hY hoccX hoccY hsep |
| v hv hhigh j hj z hz ξ |
| have hNpos : 0 < e.N := lt_of_lt_of_le Nat.zero_lt_one ((le_max_left _ _).trans he) |
| have hN₁ : N₁ ≤ e.N := (le_trans (le_max_left _ _) (le_max_right _ _)).trans he |
| have hN₂ : N₂ ≤ e.N := (le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) |
| (le_max_right _ _)).trans he |
| have hN₃ : N₃ ≤ e.N := (le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) |
| (le_max_right _ _)).trans he |
| let := hρ.probability |
| have hρ' := (_root_.OAI.CurrentFalconer.TreeND.literal_regularizedProfileData_iff e.ρ e.N B s sig e.mass).mp hρ |
| have hinterior := policyDepth_interior (adjustedProfile (d:=n+1) e.mass e.N) hq |
| (policyBlocks_nonempty hq ht hwide) |
| have hrange := prefix_high_graph_ranges hsig hq hNpos hinterior hwide hqscale hprefix ht hv hhigh |
| let L : ℝ := (((e.t-e.a)/2:ℕ):ℝ) |
| have hb := e.prefixPhased_jets (m:=m) |
| (S:=((2:ℝ)^e.N)^sig*max 1 ((2:ℝ)^L*((2:ℝ)^e.N)^(-50*sig))) |
| (by positivity : 0 ≤ D^(2*Order)) z ξ |
| (fun x r hr w hw => by |
| have hh := (hder e.N hN₂ e rfl hse hq j hj z hz x r hr w hw).1 |
| simpa only [sourceMaskScaleND,Real.rpow_natCast,L] using hh) |
| (fun x r hr w hw => by |
| have hh := (hder e.N hN₂ e rfl hse hq j hj z hz x r hr w hw).2 |
| simpa only [sourceMaskScaleND,Real.rpow_natCast,L] using hh) |
| have htest := himpose e hN₃ hse hJe inferInstance hq ht hwide hprefix z ξ |
| have htest₁ : _root_.OAI.CurrentFalconer.TreeND.ImposesParallelTest e.ρ e.a e.sourcePrefixDepth |
| (_root_.OAI.CurrentFalconer.TreeND.testWidth e.N Jsmooth (e.i+1) e.sourcePrefixDepth sig) |
| (((2:ℝ)^e.N)^(_root_.OAI.PlanarFalconer.height |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N) |
| e.a e.sourcePrefixDepth+e.C*sig)) (e.prefixPhasedLeft z ξ) := by |
| convert htest.1 using 1 |
| rfl |
| have htest₂ : _root_.OAI.CurrentFalconer.TreeND.ImposesParallelTest e.ρ e.a e.sourcePrefixDepth |
| (_root_.OAI.CurrentFalconer.TreeND.testWidth e.N Jsmooth (e.i+1) e.sourcePrefixDepth sig) |
| (((2:ℝ)^e.N)^(_root_.OAI.PlanarFalconer.height |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N) |
| e.a e.sourcePrefixDepth+e.C*sig)) (e.prefixPhasedRight z ξ) := by |
| convert htest.2 using 1 |
| rfl |
| have hX' : e.X ∈ _root_.OAI.CurrentFalconer.TreeND.dyadicPaths e.a := |
| (_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths _ _).mpr hX |
| have hY' : e.Y ∈ _root_.OAI.CurrentFalconer.TreeND.dyadicPaths e.a := |
| (_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths _ _).mpr hY |
| have hoccX' : 0 < e.ρ (_root_.OAI.CurrentFalconer.TreeND.treeCell e.X) := by |
| simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hoccX |
| have hoccY' : 0 < e.ρ (_root_.OAI.CurrentFalconer.TreeND.treeCell e.Y) := by |
| simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hoccY |
| have hi' : e.i+1 ≤ Jsmooth := by omega |
| have hanchor : (e.a:ℝ) ≤ (e.a:ℝ)+2*sig*e.N := le_add_of_nonneg_right (by positivity) |
| have hh := hgraph e.N hN₁ e.ρ e.ρ B s s e.mass e.mass hρ' hρ' |
| e.a e.sourcePrefixDepth v L e.C BΨ (D^(2*Order)) Pcut Jsmooth (e.i+1) (e.i+1) |
| e.a e.a e.X e.Y Ψ e.prefixCutoff (e.prefixPhasedLeft z ξ) (e.prefixPhasedRight z ξ) |
| hrange.1 hrange.2.1 hrange.2.2.1 hrange.2.2.2.1 hrange.2.2.2.2.1 hrange.2.2.2.2.2 |
| le_rfl le_rfl hrange.1 hrange.1 hanchor hanchor hJ hi' hi' |
| hX' hY' hoccX' hoccY' (e.prefix_graph_centers hX hY hsep) |
| hΨ (hsupp.trans Ioo_subset_Icc_self) hΨ0 hintΨ (hderivΨ m le_rfl) hBΨ |
| hb.1 hb.2 hPcut e.prefixCutoff_contDiff (fun w _ => e.prefixCutoff_norm_le w) |
| (fun j hj w hw => (hcut e hsep j w hw).trans (hPbound j hj)) htest₁ htest₂ |
| rw [←e.prefixAngularHF_graph_eq] at hh |
| have hfactor : ((2:ℝ)^e.N)^(-( |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N e.sourcePrefixDepth- |
| _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N e.a)+ |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N e.sourcePrefixDepth- |
| _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N e.a))+ |
| min (_root_.OAI.PlanarFalconer.height |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N) e.a e.sourcePrefixDepth) |
| (_root_.OAI.PlanarFalconer.height |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N) e.a e.sourcePrefixDepth)+ |
| (e.C+36*(n+1:ℝ)+5)*sig) = e.prefixGraphFactor := by |
| unfold PositiveDistanceState.prefixGraphFactor |
| rw [hse,min_self] |
| congr 1 |
| change -((adjustedProfile (d:=n+1) e.mass e.N e.sourcePrefixDepth- |
| adjustedProfile (d:=n+1) e.mass e.N e.a)+(adjustedProfile (d:=n+1) e.mass e.N e.sourcePrefixDepth- |
| adjustedProfile (d:=n+1) e.mass e.N e.a))+height (adjustedProfile (d:=n+1) e.mass e.N) |
| e.a e.sourcePrefixDepth+(e.C+36*(n+1:ℝ)+5)*sig = _ |
| ring |
| simpa only [Nat.cast_add,Nat.cast_one,hfactor,PositiveDistanceState.prefixGraphChildMax] |
| using hh |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND.PositiveDistanceState |
| open Set MeasureTheory |
| variable {n : ℕ} (e : PositiveDistanceState (n+1)) |
|
|
| noncomputable def prefixHighChildValues (Order : ℕ) (v : ℝ) (Ψ : ℝ → ℝ) : Set ℝ := |
| insert 0 {r | ∃ j ≤ Order, ∃ z ∈ Attempt4PrefixShell.inverseAngularTermsND j |
| (e.prefixDescriptors.disjSum e.prefixDescriptors), |
| ∃ ξ : EuclideanSpace ℝ (Fin (n+1)), r=e.prefixGraphChildMax v Ψ z ξ} |
| noncomputable def prefixHighChildSup (Order : ℕ) (v : ℝ) (Ψ : ℝ → ℝ) : ℝ := |
| sSup (e.prefixHighChildValues Order v Ψ) |
| end GraphLiteralSupport.CurrentFalconer.TreeND.PositiveDistanceState |
| end |
| end |
|
|
| end OAI |
|
|
