openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphSourceFourSpatialDecay.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphSourceFixedGeometry | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SpherePlaneWaveSub | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| namespace CurrentFalconer | |
| open Set MeasureTheory Attempt4FullFar | |
| open scoped ComplexConjugate RealInnerProductSpace ContDiff | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| theorem graph_source_four_spatial_decay | |
| {sig c₀ Ccap G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) | |
| (hCcap : 0 ≤ Ccap) (hG : 0 ≤ G) (T : ℝ) : | |
| ∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (a p v L : ℝ), p-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p → | |
| let R := (2:ℝ)^N | |
| let δ := min c₀ (((2:ℝ)^(p-a))⁻¹*R^(10*sig)) | |
| let τ := (2:ℝ)^(-p)*R^(30*sig) | |
| let A := 1+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹ | |
| ∀ (μ₁ μ₂ μ₃ μ₄ : Measure V), | |
| ∀ [IsFiniteMeasure μ₁] [IsFiniteMeasure μ₂] [IsFiniteMeasure μ₃] [IsFiniteMeasure μ₄], | |
| ∀ (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ : V) (K J₁ J₂ : ℝ) | |
| (θ₁ θ₂ : V → ℂ) (b₁ b₂ b₃ b₄ : V → V → ℂ) | |
| (P₁ P₂ B₁ B₂ B₃ B₄ : ℝ), | |
| ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → | |
| (∫ u : ℝ, ‖Ψ u‖) ≤ K → (∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) → | |
| ‖w₁‖=1 → ‖w₂‖=1 → 0 ≤ K → 0 ≤ J₁ → 0 ≤ J₂ → | |
| Measurable θ₁ → Measurable θ₂ → | |
| Measurable (Function.uncurry b₁) → Measurable (Function.uncurry b₂) → | |
| Measurable (Function.uncurry b₃) → Measurable (Function.uncurry b₄) → | |
| 0 ≤ P₁ → 0 ≤ P₂ → 0 ≤ B₁ → 0 ≤ B₂ → 0 ≤ B₃ → 0 ≤ B₄ → | |
| (∀ w, ‖w‖=1 → ‖θ₁ w‖ ≤ P₁) → (∀ w, ‖w‖=1 → ‖θ₂ w‖ ≤ P₂) → | |
| (∀ x w, ‖w‖=1 → ‖b₁ x w‖ ≤ B₁) → (∀ x w, ‖w‖=1 → ‖b₂ x w‖ ≤ B₂) → | |
| (∀ x w, ‖w‖=1 → ‖b₃ x w‖ ≤ B₃) → (∀ x w, ‖w‖=1 → ‖b₄ x w‖ ≤ B₄) → | |
| (∀ᵐ z ∂(μ₁.prod μ₂).prod (μ₃.prod μ₄), | |
| ‖z.1.1-z.1.2‖ ≤ G*(2:ℝ)^(-a) ∧ ‖z.2.1-z.2.2‖ ≤ G*(2:ℝ)^(-a) ∧ | |
| ‖(z.1.1-z.1.2)-d₁‖ ≤ G*(2:ℝ)^(-p) ∧ ‖(z.2.1-z.2.2)-d₂‖ ≤ G*(2:ℝ)^(-p)) → | |
| ¬(‖d₁-inner ℝ d₁ w₁ • w₁‖ ≤ τ ∧ | |
| ‖d₂-inner ℝ d₂ w₂ • w₂‖ ≤ τ ∧ |inner ℝ w₁ d₁+inner ℝ w₂ d₂| ≤ τ) → | |
| (∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ₁ b₁ b₂ z)) → | |
| (∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ₂ b₃ b₄ z)) → | |
| (∀ z j, j ≤ m → ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ j (graphPairAmplitude θ₁ b₁ b₂ z) w‖ ≤ J₁*A^j) → | |
| (∀ z j, j ≤ m → ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ j (graphPairAmplitude θ₂ b₃ b₄ z) w‖ ≤ J₂*A^j) → | |
| (∀ w, ‖w‖=1 → θ₁ w ≠ 0 → ‖w-w₁‖ ≤ Ccap*δ) → | |
| (∀ w, ‖w‖=1 → θ₂ w ≠ 0 → ‖w-w₂‖ ≤ Ccap*δ) → | |
| ‖∫ u, Ψ u*graphPairMoment μ₁ μ₂ b₁ b₂ θ₁ ((2:ℝ)^v*u)* | |
| graphPairMoment μ₃ μ₄ b₃ b₄ θ₂ ((2:ℝ)^v*u)‖ ≤ | |
| C*K*J₁*J₂*R^(-T)*((μ₁.prod μ₂).prod (μ₃.prod μ₄)).real univ := by | |
| obtain ⟨m,C,hC,N₀,hN₀⟩ := graph_source_fixed_nonedge_decay V hsig hc₀ hCcap hG T | |
| refine ⟨m,C,hC,N₀,?_⟩ | |
| intro N hN a p v L hh hL hq R δ τ A μ₁ μ₂ μ₃ μ₄ _ _ _ _ | |
| Ψ w₁ w₂ d₁ d₂ K J₁ J₂ θ₁ θ₂ b₁ b₂ b₃ b₄ P₁ P₂ B₁ B₂ B₃ B₄ | |
| hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hJ₁ hJ₂ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄ | |
| hP₁ hP₂ hB₁ hB₂ hB₃ hB₄ hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄ hgeo hne | |
| ha₁ ha₂ hd₁ hd₂ hs₁ hs₂ | |
| have hiΨ : Integrable Ψ := hΨ.continuous.integrable_of_hasCompactSupport | |
| (isCompact_Icc.of_isClosed_subset (isClosed_tsupport Ψ) hsΨ) | |
| rw [graph_four_spatial_factorization μ₁ μ₂ μ₃ μ₄ hΨ.continuous.measurable hiΨ ((2:ℝ)^v) θ₁ θ₂ b₁ b₂ b₃ b₄ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄ hP₁ hP₂ hB₁ hB₂ hB₃ hB₄ hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄] | |
| apply norm_integral_le_of_norm_le_const | |
| filter_upwards [hgeo] with z hz | |
| apply hN₀ N hN a p v L hh hL hq Ψ w₁ w₂ d₁ d₂ | |
| (z.1.1-z.1.2) (z.2.1-z.2.2) K J₁ J₂ | |
| (graphPairAmplitude θ₁ b₁ b₂ z.1) (graphPairAmplitude θ₂ b₃ b₄ z.2) | |
| hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hJ₁ hJ₂ hz.1 hz.2.1 hz.2.2.1 hz.2.2.2 hne | |
| (ha₁ z.1) (ha₂ z.2) (hd₁ z.1) (hd₂ z.2) | |
| · intro w hw hne | |
| apply hs₁ w hw | |
| intro hz | |
| apply hne | |
| simp [graphPairAmplitude,hz] | |
| · intro w hw hne | |
| apply hs₂ w hw | |
| intro hz | |
| apply hne | |
| simp [graphPairAmplitude,hz] | |
| end CurrentFalconer | |
| end | |
| end | |
| end OAI | |