File size: 6,905 Bytes
caec230 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | 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
end OAI
|