openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Certificates /ContinuousDiagonalLinearStep.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusAssembly | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.DiagonalRealCoordinates | |
| namespace OAI | |
| /-! # Continuous normalized linear data in the real diagonal symmetry coordinates -/ | |
| open scoped SchwartzMap NNReal | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| def HasContinuousFiniteDiagonalTorusLinearSteps {H V : Type*} | |
| [NormedAddCommGroup H] [NormedSpace ℝ H] | |
| [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V] | |
| (π : H →L[ℝ] V) (A : ℝ≥0 → Radius → FourierL2 →L[ℝ] FourierL2) : Prop := | |
| ∃ G : V →L[ℂ] V, | |
| ∃ hspan : (⨆ lam : ℂ, Module.End.eigenspace G.toLinearMap lam) = ⊤, | |
| ∃ hspec : ∀ (lam : ℂ) (v : V), v ≠ 0 → G v = lam • v → | |
| lam = 0 ∨ lam = 1 ∨ lam = 1 / 2, | |
| ∃ T₀ : ℝ, 0 ≤ T₀ ∧ ∀ T : ℝ≥0, T₀ ≤ T → | |
| HasContinuousTorusLinearStep (F := (diagonalRealCoordinates π G hspan hspec).range) T (A T) | |
| theorem continuous_finiteDiagonalTorusLinearSteps_of_transfer {V : Type*} | |
| [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V] | |
| (a k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ)) | |
| (ρ : ℝ) (hρ : 0 < ρ) (hχ : ∀ y : E, ‖y‖ ≤ ρ → χ y = 1) | |
| (π : HomogeneousY a k →L[ℝ] V) (G : V →L[ℂ] V) | |
| (hspan : (⨆ lam : ℂ, Module.End.eigenspace G.toLinearMap lam) = ⊤) | |
| (hspec : ∀ (lam : ℂ) (v : V), v ≠ 0 → G v = lam • v → | |
| lam = 0 ∨ lam = 1 ∨ lam = 1 / 2) | |
| (A : ℝ≥0 → Radius → FourierL2 →L[ℝ] FourierL2) | |
| (hkernel : ∀ β : ℝ, 0 < β → ∃ T₀ : ℝ, 0 ≤ T₀ ∧ | |
| ∀ T : ℝ≥0, T₀ ≤ T → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 → | |
| ∀ f : FourierL2, ‖f‖ ≤ 1 → | |
| π (homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ f) = 0 → ‖A T L f‖ < β) | |
| (hcoord : ∀ T, HasContractingTorusCoordinateTransfer a k ha ha1 hk χ | |
| (diagonalRealCoordinates π G hspan hspec) T (A T)) | |
| (hA : ∀ T, ∃ C : ℝ, 0 ≤ C ∧ ∀ L, ‖A T L‖ ≤ C) : | |
| HasContinuousFiniteDiagonalTorusLinearSteps π A := by | |
| let : FiniteDimensional ℝ (SymmetryCoordinates G) := | |
| FiniteDimensional.trans ℝ ℂ (SymmetryCoordinates G) | |
| refine ⟨G, hspan, hspec, ?_⟩ | |
| apply continuous_torusLinearSteps_of_transfer a k ha ha1 hk χ ρ hρ hχ | |
| (diagonalRealCoordinates π G hspan hspec) A | |
| · intro β hβ | |
| obtain ⟨T₀, hT₀, hT⟩ := hkernel β hβ | |
| refine ⟨T₀, hT₀, fun T hTT => ?_⟩ | |
| obtain ⟨L₀, hL₀⟩ := hT T hTT | |
| refine ⟨L₀, fun L hL f hf hz => ?_⟩ | |
| exact hL₀ L hL f hf ((diagonalRealCoordinates_zero_iff π G hspan hspec _).mp hz) | |
| · exact hcoord | |
| · exact hA | |
| end DefocusingNLS | |
| end OAI | |