| import OAI.MeasureTheory.Falconer.Graphs.ActualProductEnergyFinite |
| import OAI.MeasureTheory.Falconer.Angular.ActualAngularDropping |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ENNReal NNReal ContDiff |
|
|
| theorem finiteEnergyMaximum_le_mul {ι : Type*} (S : Finset ι) (E F : ι → ℝ≥0∞) |
| {c : ℝ} (hc : 0 ≤ c) (hb : ∀ i ∈ S, (E i).toReal ≤ c*(F i).toReal) : |
| finiteEnergyMaximum S E ≤ c*finiteEnergyMaximum S F := by |
| classical |
| by_cases hs : S.Nonempty |
| · obtain ⟨i,hi,he⟩ := finiteEnergyMaximum_attained S hs E |
| rw [he] |
| exact (hb i hi).trans (mul_le_mul_of_nonneg_left (le_finiteEnergyMaximum S F hi) hc) |
| · have he : S = ∅ := Finset.not_nonempty_iff_eq_empty.mp hs |
| simp [he,finiteEnergyMaximum_empty] |
|
|
| noncomputable def remainingAngularEnergy |
| {ι : Fin 2 → Type} (S : ∀ l, Finset (ι l)) (d : ∀ l, ι l → PolicyDescriptor) |
| (k : ∀ l, ι l → ℕ) (Q : Fin 2 → DyadicSquare) (ρ : Fin 2 → Measure Plane) |
| (mass : Fin 2 → ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) (h : Fin 2 → Plane → ℂ) |
| (M₁ M₂ p v : ℝ) (Ψ : ℝ → ℝ) (row : CellPath × CellPath) : ℝ≥0∞ := |
| sourceAngularH p v Ψ |
| (fun z => sourceAngularTransform (ρ 0) ((Q 0).descendant row.1).carrier M₁ |
| (inheritedCircleSymbol ((S 0).filter (fun c => row.1.length < descriptorAnchor (d 0 c))) |
| (d 0) (k 0) (Q 0) (ρ 0) (mass 0) N lam sig J i (h 0)) z.1 z.2) |
| (fun z => sourceAngularTransform (ρ 1) ((Q 1).descendant row.2).carrier M₂ |
| (inheritedCircleSymbol ((S 1).filter (fun c => row.2.length < descriptorAnchor (d 1 c))) |
| (d 1) (k 1) (Q 1) (ρ 1) (mass 1) N lam sig J i (h 1)) z.1 z.2) |
|
|
| theorem inherited_product_energy_drop_real |
| {ι : Fin 2 → Type} [∀ l, DecidableEq (ι l)] |
| (S : ∀ l, Finset (ι l)) (d : ∀ l, ι l → PolicyDescriptor) |
| (k : ∀ l, ι l → ℕ) (K : ℕ) (hK : ∀ l, (∑ t ∈ S l, k l t) ≤ K) |
| (Q : Fin 2 → DyadicSquare) (hQ : ∀ l, 0 < (Q l).side) |
| (ρ : Fin 2 → Measure Plane) [∀ l, IsFiniteMeasure (ρ l)] |
| (mass : Fin 2 → ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) |
| (h : Fin 2 → Plane → ℂ) (hh : ∀ l, Measurable (h l)) (hhb : ∀ l x, ‖h l x‖ ≤ 1) |
| (row : CellPath × CellPath) (p v : ℝ) {M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂) |
| {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hpos : ∀ t, 0 ≤ Ψ t) |
| (hs : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) : |
| let β := fun l => inheritedCircleSymbol (S l) (d l) (k l) (Q l) (ρ l) (mass l) N lam sig J i (h l) |
| let E := sourceAngularH p v Ψ |
| (fun z => sourceAngularTransform (ρ 0) ((Q 0).descendant row.1).carrier M₁ (β 0) z.1 z.2) |
| (fun z => sourceAngularTransform (ρ 1) ((Q 1).descendant row.2).carrier M₂ (β 1) z.1 z.2) |
| let F := remainingAngularEnergy S d k Q ρ mass N lam sig J i h M₁ M₂ p v Ψ row |
| F ≠ ⊤ ∧ E.toReal ≤ (symbolDerivativeConstant K^K)^4*F.toReal := by |
| classical |
| let P : Fin 2 → CellPath := ![row.1,row.2] |
| let S' := fun l => (S l).filter (fun c => (P l).length < descriptorAnchor (d l c)) |
| have hb (l : Fin 2) : (∑ t ∈ S' l, k l t) ≤ K := |
| remaining_symbol_budget (S l) (d l) (k l) (P l).length K (hK l) |
| have hf := inherited_product_energy_finite S' d k K hb Q hQ ρ mass N lam sig J i h hh hhb |
| ((Q 0).descendant row.1).carrier ((Q 1).descendant row.2).carrier p v hM₁ hM₂ hΨ hpos hs |
| have hdrop := actual_inherited_H_drop (S 0) (S 1) (d 0) (d 1) (k 0) (k 1) (Q 0) (Q 1) |
| (hQ 0) (hQ 1) (ρ 0) (ρ 1) (mass 0) (mass 1) N lam sig J i row.1 row.2 (h 0) (h 1) |
| K K (hK 0) (hK 1) |
| (((Q 0).descendant row.1).center_mem ((Q 0).descendant_side_pos (hQ 0) row.1)) |
| (((Q 1).descendant row.2).center_mem ((Q 1).descendant_side_pos (hQ 1) row.2)) M₁ M₂ p v Ψ |
| dsimp only |
| have hf' : remainingAngularEnergy S d k Q ρ mass N lam sig J i h M₁ M₂ p v Ψ row ≠ ⊤ := hf |
| refine ⟨hf',?_⟩ |
| have hn : 0 ≤ symbolDerivativeConstant K^K := pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le K)) K |
| have ht := ENNReal.toReal_mono |
| (ENNReal.mul_ne_top (ENNReal.mul_ne_top (ENNReal.pow_ne_top ENNReal.ofReal_ne_top) |
| (ENNReal.pow_ne_top ENNReal.ofReal_ne_top)) hf') hdrop |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_pow,ENNReal.toReal_ofReal hn] at ht |
| convert ht using 1 |
| ring |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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