openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingCompactLowProduct.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingLowMassLimit | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactMassLimit | |
| namespace OAI | |
| /-! # Compact lower-order product for the actual unfiltered field -/ | |
| open Filter Topology | |
| open scoped SchwartzMap | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem tendsto_expandingSample_lowEnergy_product (a M R : ℝ) (N : ℕ) | |
| (ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ)) | |
| (L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop) | |
| (f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M) | |
| (hlocal : ∀ R ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ y : E, ‖y‖ ≤ R → | |
| ‖expandingTorusFunction a N (L n) (f n) (euclideanToTorus ((L n)⁻¹ • y))‖ < ε) | |
| (K : 𝓢(E, ℂ)) (hK : ∀ y : E, R < ‖y‖ → K y = 0) : | |
| Tendsto (fun n => expandingLowEnergy a N (L n) (hL n) | |
| (expandingProduct a N (L n) ha ha1 hN (hL n) | |
| (schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K)) (f n))) | |
| atTop (𝓝 0) := by | |
| obtain ⟨C, hC, hc⟩ := exists_schwartzTorusSample_norm_bound a N ha1 hN (radianFourierKernel K) | |
| let A : ℝ := 2 * Real.sqrt (4 ^ (N : ℝ)) * expandingEmbeddingBound a N | |
| have hA : 0 ≤ A := by dsimp [A]; unfold expandingEmbeddingBound; positivity | |
| let g := fun n => expandingProduct a N (L n) ha ha1 hN (hL n) | |
| (schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K)) (f n) | |
| have hg (n : ℕ) : ‖g n‖ ≤ A * C * M := | |
| (expandingProduct_norm_le a N (L n) ha ha1 hN (hL n) _ _).trans | |
| (mul_le_mul (mul_le_mul_of_nonneg_left (hc (L n) (hL n)) hA) | |
| (hf n) (norm_nonneg _) (mul_nonneg hA hC)) | |
| exact tendsto_expandingLowEnergy_of_mass a (A * C * M) N ha ha1 hN L hL g hg | |
| (tendsto_expandingSample_product_mass a N M R ha ha1 hN L hL hLinf f hf hlocal K hK) | |
| end DefocusingNLS | |
| end OAI | |