openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Shells /ActualComparableShellDistance.lean
| import OAI.MeasureTheory.Falconer.Shells.ActualComparableShellRebin | |
| import OAI.MeasureTheory.Falconer.Geometry.SeparatedDistanceMaximum | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open Set MeasureTheory | |
| open scoped ENNReal ContDiff | |
| attribute [local instance] Classical.propDecidable | |
| theorem uniform_actual_comparable_shell_distance {Ψ : ℝ → ℝ} | |
| (hΨ : ContDiff ℝ ∞ Ψ) (hsΨ : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) | |
| {sig : ℝ} (hsig : 0 < sig) (K o : ℕ) {J : ℕ} (hJ : 0 < J) : | |
| ∃ H : ℝ, 0 ≤ H ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (Q : Fin 2 → DyadicSquare), (∀ l, (Q l).side = 1) → | |
| ∀ (ρ : Fin 2 → Measure Plane) [∀ l, IsFiniteMeasure (ρ l)], | |
| (∀ l, ρ l (Q l).carrierᶜ = 0) → | |
| ∀ (m : ℝ → ℝ) (q a₀ t₀ a d n v : ℕ), | |
| 0 < q → a₀ ≤ a+n → v ≤ t₀ → v ≤ N → | |
| ∀ (S : Fin 2 → Finset PolicyDescriptor) (k : Fin 2 → PolicyDescriptor → ℕ), | |
| (∀ l, remainingPolicyDescriptors (fun t => m t-(t:ℝ)/(N:ℝ)) N q a₀ t₀ a ⊆ S l) → | |
| (∀ l, (S l).card ≤ (N+1)^3+1) → (∀ l, ∑ t ∈ S l, k l t ≤ K) → | |
| ∀ L : ℕ, (∀ l t, t ∈ S l → descriptorLength t ≤ L) → | |
| ∀ L₀ : ℝ, (v:ℝ)-(a+d:ℕ) = 2*L₀ → (L:ℝ) ≤ L₀ → 10*sig*(N:ℝ) ≤ L₀ → | |
| ∀ (lam : ℝ) (i : Fin 2 → ℕ) (hsp : Fin 2 → Plane → ℂ), | |
| (∀ l x, ‖hsp l x‖ ≤ 1) → | |
| ∀ A B : CellPath, A.length = a → B.length = a → | |
| let β := fun l => inheritedCircleSymbol (S l) id (k l) (Q l) (ρ l) (m) | |
| N (lam) sig J (i l) (hsp l) | |
| let D : Fin 2 → PositiveDistanceData := fun l => | |
| ⟨Q l,ρ l,m,N,q,a+n,v,lam,sig,J,i l+1,[],[]⟩ | |
| ‖∑ row ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B d, | |
| ∑ col ∈ sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B d, | |
| if ¬ angularNear (Q 0) (Q 1) (a+d) row col then | |
| ∫ x in angularLengthShell (a+n), angularStationarySpatialKernel (β 0) (β 1) o v Ψ x | |
| ∂angularFourCellMeasure (Q 0) (Q 1) (ρ 0) (ρ 1) row col | |
| else 0‖ₑ ≤ | |
| ENNReal.ofReal H * binKernelConstant * | |
| ENNReal.ofReal ((((2:ℝ)^N)^(-m (a+n)))^4*(2:ℝ)^(-2*((v:ℝ)-(a+n:ℕ)))) * | |
| ((sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n).card:ℝ≥0∞) * | |
| 257025537024256 * | |
| (D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n) := by | |
| obtain ⟨H,hH,N₀,h₀⟩ := uniform_actual_comparable_shell_rebin hΨ hsΨ hsig K o hJ | |
| refine ⟨H,hH,N₀,?_⟩ | |
| intro N hN Q hQ ρ inst hcarry m q a₀ t₀ a d n v hq ha₀ ht hv S k hS hcard hK L hL | |
| L₀ hvh hLL hL₀ lam i hsp hspb A B hA hB | |
| let D : Fin 2 → PositiveDistanceData := fun l => | |
| ⟨Q l,ρ l,m,N,q,a+n,v,lam,sig,J,i l+1,[],[]⟩ | |
| have he := h₀ N hN Q hQ ρ hcarry (fun _ => m) q a₀ t₀ a d n v hq ha₀ ht hv | |
| S k hS hcard hK L hL L₀ hvh hLL hL₀ (fun _ => lam) i hsp hspb A B hA hB | |
| have hn : (D 0).normalization = (D 1).normalization := rfl | |
| have hk := PositiveDistanceData.rebinned_positive_kernel_max (D 0) (D 1) rfl hn A B n (a+n) | |
| have hc := PositiveDistanceData.separated_sum_le_maximum (D 0) (D 1) (hQ 0) (hQ 1) (a+n) | |
| (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) | |
| (sourceProductVertices_subset _ _ _ _ _ _ a n hA hB) | |
| dsimp only at he ⊢ | |
| rw [mul_assoc] at he | |
| have hh := he.trans (mul_le_mul_right hk (ENNReal.ofReal H)) | |
| have hh' := hh.trans (mul_le_mul_right | |
| (mul_le_mul_right hc (binKernelConstant * ENNReal.ofReal | |
| ((D 0).levelMass^4*(2:ℝ)^(-2*((v:ℝ)-(a+n:ℕ)))))) (ENNReal.ofReal H)) | |
| simpa only [D,PositiveDistanceData.levelMass,mul_assoc,Nat.cast_add] using hh' | |
| end PlanarFalconer | |
| end OAI | |