| import OAI.MeasureTheory.Falconer.Recursion.RecursiveCellInvariants |
| import OAI.MeasureTheory.Falconer.Recursion.BranchExponent |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ContDiff |
|
|
| theorem AngularStateData.mul_childMaximum_le (d : AngularStateData) |
| (k : Fin 2 → PolicyDescriptor → ℕ) (h : Fin 2 → Plane → ℂ) |
| (p : ℕ) (Ψ : ℝ → ℝ) {c M : ℝ} (hc : 0 < c) (hM : 0 ≤ M) |
| (hb : ∀ row ∈ d.children p, c*((d.successor p row).energy k h Ψ).toReal ≤ M) : |
| c*d.childMaximum k h p Ψ ≤ M := by |
| have hh := d.childMaximum_le k h p Ψ (div_nonneg hM hc.le) |
| (fun row hr => (le_div_iff₀ hc).mpr (by simpa only [mul_comm] using hb row hr)) |
| exact (mul_le_mul_of_nonneg_left hh hc.le).trans_eq (by field_simp) |
|
|
| theorem AngularStateData.mul_farDistanceMaximum_le (d : AngularStateData) |
| [IsFiniteMeasure d.ρ] (b : ℕ) {c M : ℝ} (hc : 0 < c) (hM : 0 ≤ M) |
| (hb : ∀ row ∈ d.children b, ∀ col ∈ d.children b, |
| angularSeparated d.Q d.Q b row col → |
| c*((d.distanceSuccessor b).atCells row.1 col.1).coincidence.toReal ≤ M ∧ |
| c*((d.distanceSuccessor b).atCells row.2 col.2).coincidence.toReal ≤ M) : |
| c*(d.farDistanceMaximum b).toReal ≤ M := by |
| have hh := d.farDistanceMaximum_le b (div_nonneg hM hc.le) (by |
| intro row hr col hcol hs |
| have hh := hb row hr col hcol hs |
| exact ⟨(le_div_iff₀ hc).mpr (by simpa only [mul_comm] using hh.1), |
| (le_div_iff₀ hc).mpr (by simpa only [mul_comm] using hh.2)⟩) |
| exact (mul_le_mul_of_nonneg_left hh hc.le).trans_eq (by field_simp) |
|
|
| theorem uniform_actual_angular_recursion_bound {lam sig kap eps : ℝ} |
| (hlam : 0 ≤ lam) (hsig : 0 < sig) (hkap : 0 < kap) |
| (hgap : sig < kap/1000) (heps : 0 < eps) (T : ℝ) (K : ℕ) |
| {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hpos : ∀ x, 0 ≤ Ψ x) |
| (hs : ∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) : |
| ∃ N₀ : ℕ, ∀ d : AngularStateData, |
| d.lam = lam → d.sig = sig → d.q = sourcePolicyBlock d.N kap → |
| d.J = sourceSmoothingLevels sig kap → N₀ ≤ d.N → |
| d.Q.side = 1 → d.ρ d.Q.carrierᶜ = 0 → |
| d.branchState.Valid d.N d.sig → d.a+100*d.q < d.v → d.i+1 ≤ d.J → |
| d.X.length = d.a → d.Y.length = d.a → |
| ∀ B : ℕ, ∀ δ : ℝ, RegularizedProfileData d.Q d.ρ d.N B δ d.lam d.mass → |
| ∀ k : Fin 2 → PolicyDescriptor → ℕ, (∀ l, ∑ e ∈ d.retainedList d.a, k l e ≤ K) → |
| ∀ h : Fin 2 → Plane → ℂ, (∀ l, Measurable (h l)) → (∀ l x, ‖h l x‖ ≤ 1) → |
| ∀ M : ℝ, 0 ≤ M → |
| (∀ p l, BranchStep d.profile d.N d.q d.sig d.branchState l (.H d.a₀ d.t₀ p d.v d.i) → |
| ∀ row ∈ d.children p, |
| ((2:ℝ)^d.N)^(branchExponent d.profile lam sig eps d.branchState l (.H d.a₀ d.t₀ p d.v d.i)) * |
| ((d.successor p row).energy k h Ψ).toReal ≤ M) → |
| (∀ b, BranchStep d.profile d.N d.q d.sig d.branchState .far (.D b d.v (d.i+1)) → |
| ∀ row ∈ d.children b, ∀ col ∈ d.children b, angularSeparated d.Q d.Q b row col → |
| ((2:ℝ)^d.N)^(branchExponent d.profile lam sig eps d.branchState .far (.D b d.v (d.i+1))) * |
| ((d.distanceSuccessor b).atCells row.1 col.1).coincidence.toReal ≤ M ∧ |
| ((2:ℝ)^d.N)^(branchExponent d.profile lam sig eps d.branchState .far (.D b d.v (d.i+1))) * |
| ((d.distanceSuccessor b).atCells row.2 col.2).coincidence.toReal ≤ M) → |
| d.energy k h Ψ ≠ ⊤ ∧ |
| (d.energy k h Ψ).toReal ≤ ((d.N:ℝ)+1)*M+((2:ℝ)^d.N)^(-T) := by |
| obtain ⟨N₀,h₀⟩ := source_angular_reduction hlam hsig hkap hgap heps T K hΨ hpos hs |
| refine ⟨N₀,?_⟩ |
| intro d hdlam hdsig hdq hdJ hN hQ hcarry hv hw hi hX hY B δ hprof k hk h hh hhb M hM hH hD |
| change d.a₀ ≤ d.a ∧ d.a ≤ d.v ∧ d.v ≤ d.t₀ ∧ d.t₀ ≤ d.N ∧ |
| (d.t₀:ℝ) ≤ d.v+d.sig*d.N at hv |
| have ht : (d.t₀:ℝ) ≤ d.v+sig*d.N := by simpa only [hdsig] using hv.2.2.2.2 |
| obtain ⟨hfinite,p,hap,hpv,_,_,hd | hn⟩ := h₀ d hdlam hdsig hdq hdJ hN hQ hcarry |
| hv.1 hv.2.1 hv.2.2.1 hv.2.2.2.1 ht hw hi hX hY B δ hprof k hk h hh hhb |
| · obtain ⟨e,rfl,he,hbound⟩ := hd |
| have hstep : BranchStep d.profile d.N d.q d.sig d.branchState .direct |
| (.H d.a₀ d.t₀ (descriptorAnchor e) d.v d.i) := BranchStep.direct hw he |
| have hm := d.mul_childMaximum_le k h (descriptorAnchor e) Ψ |
| (by positivity : 0 < ((2:ℝ)^d.N)^(branchExponent d.profile lam sig eps d.branchState .direct |
| (.H d.a₀ d.t₀ (descriptorAnchor e) d.v d.i))) hM (hH _ _ hstep) |
| simp only [branchExponent,branchCost,BranchState.start,AngularStateData.branchState, reduceCtorEq,ite_false] at hm |
| refine ⟨hfinite,hbound.trans ?_⟩ |
| nlinarith [mul_nonneg (Nat.cast_nonneg (α := ℝ) d.N) hM] |
| · obtain ⟨rfl,hp5,_,hbound⟩ := hn |
| let p := d.v-2*sourceFarBuffer d.N ((d.retainedList d.a).sup descriptorLength) d.sig |
| have hstep : BranchStep d.profile d.N d.q d.sig d.branchState .near |
| (.H d.a₀ d.t₀ p d.v d.i) := BranchStep.near hw hap |
| have hm := d.mul_childMaximum_le k h p Ψ |
| (by positivity : 0 < ((2:ℝ)^d.N)^(branchExponent d.profile lam sig eps d.branchState .near |
| (.H d.a₀ d.t₀ p d.v d.i))) hM (hH _ _ hstep) |
| simp only [branchExponent,branchCost,BranchState.start,AngularStateData.branchState, reduceCtorEq,ite_false,add_zero] at hm |
| have hnear : ((2:ℝ)^d.N)^(-2*(d.profile p-d.profile d.a)+12*lam+eps)* |
| d.childMaximum k h p Ψ ≤ M := by |
| apply le_trans _ hm |
| apply mul_le_mul_of_nonneg_right _ (d.childMaximum_nonneg k h p Ψ) |
| exact Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) (by linarith) |
| let := hprof.probability |
| have hfar : d.farSum p ≤ (d.N:ℝ)*M := by |
| apply d.farSum_le_depth_factor p hp5 (hv.2.2.1.trans hv.2.2.2.1) hM |
| intro b hb |
| have hb' : b ∈ Finset.Icc (d.a+9) (p+6) := Finset.mem_Icc.mpr |
| ⟨(Finset.mem_Icc.mp hb).1, by have := (Finset.mem_Icc.mp hb).2; omega⟩ |
| have hstep : BranchStep d.profile d.N d.q d.sig d.branchState .far (.D b d.v (d.i+1)) := |
| BranchStep.far hw hap hb' |
| have hm := d.mul_farDistanceMaximum_le b |
| (by positivity : 0 < ((2:ℝ)^d.N)^(branchExponent d.profile lam sig eps d.branchState .far |
| (.D b d.v (d.i+1)))) hM (hD b hstep) |
| simp only [branchExponent,branchCost,BranchState.start,AngularStateData.branchState, reduceCtorEq,ite_false,add_zero] at hm |
| apply le_trans _ hm |
| apply mul_le_mul_of_nonneg_right _ ENNReal.toReal_nonneg |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) |
| simp only [hdlam,hdsig] |
| linarith |
| refine ⟨hfinite,hbound.trans ?_⟩ |
| change _ ≤ ((d.N:ℝ)+1)*M+_ |
| dsimp only [p] at hnear hfar |
| nlinarith |
|
|
| end PlanarFalconer |
|
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| end OAI |
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