openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /InverseAngularTermPaid.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AngularDerivativeStar | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.InverseAngularTerms | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| variable {n : ℕ} {α : Type*} [DecidableEq α] | |
| lemma inverseAngularTermND_paid (j : ℕ) (S : Finset α) (D : α → ℝ) {M r : ℝ} | |
| (hM : 1 ≤ M) (hr : 0 ≤ r) (hpaid : r*M^2 ≤ 1) | |
| (hD₀ : ∀ i ∈ S, 0 ≤ D i) (hD : ∀ i ∈ S, D i ≤ M) | |
| {a : ℂ × (α → List (SphereAmbientND n →L[ℝ] SphereAmbientND n))} | |
| (ha : a ∈ inverseAngularTermsND j S) : | |
| ‖(r:ℂ)^j * (a.1*((∏ i ∈ S, D i^(a.2 i).length:ℝ):ℂ))‖ ≤ ‖a.1‖ := by | |
| have hp : 0 ≤ ∏ i ∈ S, D i^(a.2 i).length := | |
| Finset.prod_nonneg (fun i hi => pow_nonneg (hD₀ i hi) _) | |
| rw [norm_mul,norm_pow,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hr, | |
| Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hp] | |
| calc | |
| r^j * (‖a.1‖*(∏ i ∈ S,D i^(a.2 i).length)) = | |
| ‖a.1‖*(r^j*(∏ i ∈ S,D i^(a.2 i).length)) := by ring | |
| _ ≤ ‖a.1‖*1 := mul_le_mul_of_nonneg_left | |
| (assignedAngularScaleND_paid S D a.2 hM hr hpaid hD₀ hD | |
| (inverseAngularTermsND_budget j S ha)) (norm_nonneg _) | |
| _ = _ := mul_one _ | |
| lemma inverseAngularTermsND_paid_cost (j : ℕ) (S : Finset α) (D : α → ℝ) {M r : ℝ} | |
| (hM : 1 ≤ M) (hr : 0 ≤ r) (hpaid : r*M^2 ≤ 1) | |
| (hD₀ : ∀ i ∈ S, 0 ≤ D i) (hD : ∀ i ∈ S, D i ≤ M) : | |
| (((inverseAngularTermsND (n := n) j S).map (fun a => | |
| ‖(r:ℂ)^j * (a.1*((∏ i ∈ S, D i^(a.2 i).length:ℝ):ℂ))‖)).sum) ≤ | |
| inverseAngularCostND n j*(max 1 (S.card:ℝ))^(2*j) := by | |
| apply le_trans _ (inverseAngularTermsND_cost j S) | |
| apply List.sum_le_sum | |
| intro a ha | |
| exact inverseAngularTermND_paid j S D hM hr hpaid hD₀ hD ha | |
| lemma inverseAngularTermsND_retention (j : ℕ) (S : Finset α) | |
| (f : α → SmoothFunctionND (SphereAmbientND n)) (D : α → ℝ) | |
| (hf : ∀ i ∈ S, Function.Even (f i).1) | |
| {a : ℂ × (α → List (SphereAmbientND n →L[ℝ] SphereAmbientND n))} | |
| (ha : a ∈ inverseAngularTermsND j S) : | |
| (∑ i ∈ S, (a.2 i).length ≤ 2*j) ∧ | |
| ∀ i ∈ S, Function.Even (normalizedAngularWordND (D i) (a.2 i) (f i)).1 ∧ | |
| tsupport (normalizedAngularWordND (D i) (a.2 i) (f i)).1 ⊆ tsupport (f i).1 := by | |
| refine ⟨inverseAngularTermsND_budget j S ha,?_⟩ | |
| intro i hi | |
| exact ⟨normalizedAngularWordND_even _ _ _ (hf i hi),normalizedAngularWordND_support _ _ _⟩ | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |