openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Shells /ActualComparableShellNormalized.lean
| import OAI.MeasureTheory.Falconer.Shells.ActualComparableShellDistance | |
| import OAI.MeasureTheory.Falconer.Measures.ComparableNormalization | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open Set MeasureTheory | |
| open scoped ENNReal ContDiff | |
| attribute [local instance] Classical.propDecidable | |
| theorem uniform_actual_comparable_shell_normalized {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 → | |
| ∀ (B₀ : ℕ) (δ lam : ℝ), | |
| (∀ l, RegularizedProfileData (Q l) (ρ l) N B₀ δ lam m) → a+n ≤ N → | |
| ∀ (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,[],[]⟩ | |
| ‖((2:ℝ)^(2*((v:ℝ)-a))/(((2:ℝ)^N)^(-m a))^4) • | |
| (∑ 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)‖ ≤ | |
| H * ((2:ℝ)^N)^(-2*(m (a+n)-(a+n:ℕ)/N-(m a-(a:ℝ)/N))+12*lam) * | |
| ((D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n)).toReal := by | |
| obtain ⟨H,hH,N₀,h₀⟩ := uniform_actual_comparable_shell_distance hΨ hsΨ hsig K o hJ | |
| let C : ℝ≥0∞ := ENNReal.ofReal H*binKernelConstant*257025537024256*4096 | |
| have hC : C ≠ ⊤ := by | |
| dsimp only [C] | |
| exact ENNReal.mul_ne_top | |
| (ENNReal.mul_ne_top (ENNReal.mul_ne_top ENNReal.ofReal_ne_top binKernelConstant_ne_top) (by norm_num)) | |
| (by norm_num) | |
| refine ⟨C.toReal,ENNReal.toReal_nonneg,max 1 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 B₀ δ lam hd hp 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 hNp : 0 < N := lt_of_lt_of_le (by norm_num : 0 < 1) ((le_max_left _ _).trans hN) | |
| have he := h₀ N ((le_max_right _ _).trans hN) Q hQ ρ 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 | |
| have hc := sourceProductVertices_profile_count (Q 0) (Q 1) (hQ 0) (hQ 1) (ρ 0) (ρ 1) | |
| N B₀ δ lam m m (hd 0) (hd 1) A B a n hA hB hp | |
| have hcc : ((sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n).card:ℝ) ≤ | |
| 4096*((2:ℝ)^N)^(2*(m (a+n)-m a)+12*lam) := by | |
| convert hc using 1 | |
| congr 2 | |
| ring | |
| have hn := comparable_normalized_bound N hNp m a (a+n:ℕ) v lam | |
| (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n).card | |
| (ENNReal.ofReal H*binKernelConstant) | |
| ((D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n)) | |
| _ (by simpa only [Nat.cast_add] using hcc) (by simpa only [Nat.cast_add] using he) | |
| have hmax := PositiveDistanceData.separatedMaximum_ne_top (D 0) (D 1) | |
| (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n) | |
| have hfin : C * ENNReal.ofReal (((2:ℝ)^N)^(-2*(m (a+n)-(a+n:ℕ)/N-(m a-(a:ℝ)/N))+12*lam)) * | |
| (D 0).separatedMaximum (D 1) (sourceProductVertices (Q 0) (Q 1) (ρ 0) (ρ 1) A B n) (a+n) ≠ ⊤ := | |
| ENNReal.mul_ne_top (ENNReal.mul_ne_top hC ENNReal.ofReal_ne_top) hmax | |
| simp only [Nat.cast_add] at hfin hn | |
| have ht := ENNReal.toReal_mono hfin hn | |
| have hrpow : 0 ≤ ((2:ℝ)^N)^(-2*(m ((a:ℝ)+n)-((a:ℝ)+n)/N-(m a-(a:ℝ)/N))+12*lam) := by positivity | |
| simpa only [← ofReal_norm,ENNReal.toReal_ofReal (norm_nonneg _),ENNReal.toReal_mul, | |
| ENNReal.toReal_ofReal hrpow,C,D,Nat.cast_add] using ht | |
| end PlanarFalconer | |
| end OAI | |