| import OAI.MeasureTheory.Falconer.Recursion.DirectPolicyWitness |
| import OAI.MeasureTheory.Falconer.Angular.AngularStateDropping |
| import OAI.MeasureTheory.Falconer.Estimates.ActualProductEquation13 |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ContDiff |
| namespace AngularStateData |
|
|
| theorem uniform_direct_graph {lam sig eps : ℝ} (hlam : 0 ≤ lam) (hsig : 0 < sig) |
| (heps : 0 < eps) (T : ℝ) (K J : ℕ) (hJ : 0 < J) |
| {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.J = J → N₀ ≤ d.N → |
| d.Q.side = 1 → 0 < d.q → d.v ≤ d.N → d.i < 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, ∑ b ∈ d.retainedList d.a, k l b ≤ K) → |
| ∀ h : Fin 2 → Plane → ℂ, (∀ l, Measurable (h l)) → (∀ l x, ‖h l x‖ ≤ 1) → |
| ∀ e : PolicyDescriptor, DirectPolicyWitness d.profile d.N d.q d.a₀ d.t₀ d.a d.v d.sig e → |
| let p := descriptorAnchor e |
| d.energy k h Ψ ≠ ⊤ ∧ (∀ row, d.childEnergy k h p Ψ row ≠ ⊤) ∧ |
| (∀ l, ∑ b ∈ d.retainedList p, k l b ≤ K) ∧ |
| (d.energy k h Ψ).toReal ≤ ((2:ℝ)^d.N)^ |
| (-2*(d.profile p-d.profile d.a)+height d.profile d.a p+4000*(lam+sig)+eps) * |
| d.childMaximum k h p Ψ + ((2:ℝ)^d.N)^(-T) := by |
| obtain ⟨N₁,h₁⟩ := uniform_actual_product_equation13 hsig Real.pi_pos |
| (by linarith [Real.pi_pos] : Real.pi ≤ 2*Real.pi) |
| K lam T hlam (half_pos heps) hJ hΨ hpos hs |
| obtain ⟨N₂,h₂⟩ := fixed_constant_absorbed (half_pos heps) ((symbolDerivativeConstant K^K)^4) |
| refine ⟨max N₁ N₂,?_⟩ |
| intro d hdLam hdSig hdJ hN hQ hq hvN hi hX hY B δ hprof k hk h hh hhb e hw |
| subst lam |
| subst sig |
| subst J |
| let := hprof.probability |
| have hQp : 0 < d.Q.side := by rw [hQ]; norm_num |
| let p := descriptorAnchor e |
| have hap : d.a ≤ p := hw.anchor_gt.le |
| have hpN : p ≤ d.N := hw.anchor_lt.le.trans hvN |
| have heq : d.a+(p-d.a) = p := Nat.add_sub_of_le hap |
| have heqR : (d.a:ℝ)+((p-d.a:ℕ):ℝ) = (p:ℝ) := by exact_mod_cast heq |
| have hcard (_l : Fin 2) : (d.retainedList d.a).card ≤ (d.N+1)^3+1 := |
| (remainingPolicyDescriptors_card d.profile hq).trans (Nat.le_succ _) |
| have hL (_l : Fin 2) (b : PolicyDescriptor) (hb : b ∈ d.retainedList d.a) : |
| descriptorLength (id b) ≤ descriptorLength e := by |
| rw [hw.maximal] |
| exact Finset.le_sup hb |
| have hdepth : ((p-d.a:ℕ):ℝ) ≤ (d.v:ℝ)-((d.a:ℝ)+((p-d.a:ℕ):ℝ))+2*d.sig*d.N := by |
| rw [heqR] |
| exact hw.depth_scale |
| have hlength : (descriptorLength e:ℝ) ≤ (d.v:ℝ)-((d.a:ℝ)+((p-d.a:ℕ):ℝ))+2*d.sig*d.N := by |
| rw [heqR] |
| exact hw.length_scale |
| have hv : 0 ≤ (d.v:ℝ)-((d.a:ℝ)+((p-d.a:ℕ):ℝ)) := by |
| rw [heqR] |
| exact sub_nonneg.mpr (Nat.cast_le.mpr hw.anchor_lt.le) |
| have hgap : ((max d.a (p-descriptorLength e):ℕ):ℝ)-d.a ≤ 2*d.sig*d.N := by |
| have hg := hw.g_slack |
| dsimp only [p] at * |
| linarith [show 0 ≤ d.sig*(d.N:ℝ) by positivity] |
| have hmain := h₁ d.N ((le_max_left _ _).trans hN) |
| (fun _ : Fin 2 => d.retainedList d.a) (fun _ => id) k hcard hk |
| (descriptorLength e) hL (fun _ => d.Q) (fun _ => hQ) (fun _ => d.ρ) d.mass B δ |
| (fun _ => hprof) d.i h hi hh hhb d.a (p-d.a) d.v (by omega) hv hdepth hlength |
| (by have htR : (d.v:ℝ) ≤ d.N := Nat.cast_le.mpr hvN; linarith [show (0:ℝ) ≤ (d.a:ℝ) from Nat.cast_nonneg _]) |
| (.parallel (max d.a (p-descriptorLength e)) p) hw.companion |
| (max d.a (p-descriptorLength e)) (by simp only [id_eq,heq]) |
| (le_max_left _ _) (by rw [heq]; exact hw.g_lt.le) hgap d.X d.Y hX hY |
| have hgraph : d.energy k h Ψ ≠ ⊤ ∧ |
| (d.energy k h Ψ).toReal ≤ ((2:ℝ)^d.N)^ |
| (-2*(d.profile p-d.profile d.a)+height d.profile d.a p+4000*(d.lam+d.sig)+eps/2) * |
| finiteEnergyMaximum (d.children p) (d.cellEnergy (d.retainedList d.a) k h p Ψ) |
| + ((2:ℝ)^d.N)^(-T) := by |
| dsimp only at hmain |
| simp only [heqR,heq] at hmain |
| exact hmain |
| have hk' (l : Fin 2) := d.child_budget hap (k l) (hk l) |
| refine ⟨hgraph.1,fun row => d.cellEnergy_finite hQp (d.retainedList p) k K hk' h hh hhb p hΨ hpos hs row, |
| hk',hgraph.2.trans ?_⟩ |
| have hdrop := d.child_maximum_drop hQp p hap hX hY k K hk h hh hhb hΨ hpos hs |
| have hconst := h₂ d.N ((le_max_right _ _).trans hN) |
| let E := -2*(d.profile p-d.profile d.a)+height d.profile d.a p+4000*(d.lam+d.sig) |
| have habs : ((2:ℝ)^d.N)^(E+eps/2)*(symbolDerivativeConstant K^K)^4 ≤ |
| ((2:ℝ)^d.N)^(E+eps) := by |
| calc |
| _ ≤ ((2:ℝ)^d.N)^(E+eps/2)*((2:ℝ)^d.N)^(eps/2) := by gcongr |
| _ = _ := by rw [← Real.rpow_add (by positivity)]; congr 1; ring |
| calc |
| _ ≤ ((2:ℝ)^d.N)^(E+eps/2)*((symbolDerivativeConstant K^K)^4*d.childMaximum k h p Ψ) |
| + ((2:ℝ)^d.N)^(-T) := by gcongr |
| _ = (((2:ℝ)^d.N)^(E+eps/2)*(symbolDerivativeConstant K^K)^4)*d.childMaximum k h p Ψ |
| + ((2:ℝ)^d.N)^(-T) := by ring |
| _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right habs (d.childMaximum_nonneg k h p Ψ)) le_rfl |
|
|
| end AngularStateData |
|
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| end PlanarFalconer |
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| end OAI |
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