openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalConvolutionPointwise.lean
| import OAI.Analysis.IrrationalRotation.Model.ConditionalConvolution | |
| import OAI.Analysis.IrrationalRotation.Analysis.PointwiseConvolution | |
| namespace OAI | |
| open scoped ComplexConjugate ENNReal lp | |
| namespace InvariantProjection.ConditionalConvolution | |
| theorem tsum_omittedEvent {Ω : Type*} (H : ℤ → Set Ω) (K n : ℕ) | |
| (ω : Ω) (w : ℤ → ℝ) : | |
| (∑' v : ↑(((window K n) : Set ℤ)ᶜ), | |
| (H (selector K n v)).indicator (fun _ => (1 : ℝ)) ω * w v) = | |
| ∑' v : ℤ, (omittedEvent H K n v).indicator (fun _ => (1 : ℝ)) ω * w v := by | |
| classical | |
| rw [tsum_subtype (((window K n) : Set ℤ)ᶜ) | |
| (fun v : ℤ => (H (selector K n v)).indicator (fun _ => (1 : ℝ)) ω * w v)] | |
| apply tsum_congr | |
| intro v | |
| by_cases hv : v ∈ window K n <;> simp [omittedEvent, hv] | |
| theorem pointwise_bound {Ω : Type*} (K n : ℕ) (Fx Fy : ℤ → ℝ) | |
| (ξ γ : ℓ²(ℤ, ℂ)) (hξ : ‖ξ‖ ≤ 1) (hγ : ‖γ‖ ≤ 1) | |
| (η : ℝ) (hη : 0 ≤ η) (H₁ H₂ : ℤ → Set Ω) (ω : Ω) | |
| (hratio : ∀ v : ℤ, 0 < Fy ((n : ℤ) + v) / Fx v ∧ | |
| Fy ((n : ℤ) + v) / Fx v ≤ min (Fy v / Fx v) | |
| (Fy ((n : ℤ) + v) / Fx ((n : ℤ) + v)) ∧ | |
| min (Fy v / Fx v) (Fy ((n : ℤ) + v) / Fx ((n : ℤ) + v)) ≤ 1) | |
| (hzero : (∑' v : ℤ, (ξ v / (Fx v : ℂ)) * | |
| ((Fy ((n : ℤ) + v) : ℂ) * conj (γ ((n : ℤ) + v)))) = 0) | |
| (htail : ∀ k : ℤ, (K : ℤ) < |k| → Fy k / Fx k ≤ η + | |
| (H₁ k).indicator (fun _ => (1 : ℝ)) ω + | |
| (H₂ k).indicator (fun _ => (1 : ℝ)) ω) : | |
| ‖truncated K n (fun v => ξ v / (Fx v : ℂ)) | |
| (fun w => (Fy (-w) : ℂ) * conj (γ (-w)))‖ ≤ η + | |
| (∑' v : ℤ, (omittedEvent H₁ K n v).indicator (fun _ => (1 : ℝ)) ω * | |
| (‖ξ v‖ * ‖γ ((n : ℤ) + v)‖)) + | |
| (∑' v : ℤ, (omittedEvent H₂ K n v).indicator (fun _ => (1 : ℝ)) ω * | |
| (‖ξ v‖ * ‖γ ((n : ℤ) + v)‖)) := by | |
| classical | |
| have hmask (H : ℤ → Set Ω) (k : ℤ) : | |
| ({k | ω ∈ H k} : Set ℤ).indicator (fun _ => (1 : ℝ)) k = | |
| (H k).indicator (fun _ => (1 : ℝ)) ω := by | |
| simp only [Set.indicator_apply, Set.mem_ofPred_eq] | |
| have hdom := InvariantProjection.l2_masked_convolution_tail_selector | |
| (window K n) ξ γ (Equiv.addLeft (n : ℤ)) hξ hγ | |
| (fun v => Fy ((n : ℤ) + v) / Fx v) | |
| (fun v => ⟨(hratio v).1.le, (hratio v).2.1.trans (hratio v).2.2⟩) | |
| η hη {k | ω ∈ H₁ k} {k | ω ∈ H₂ k} (selector K n) | |
| (by | |
| change (∑' v : ℤ, ((Fy ((n : ℤ) + v) / Fx v : ℝ) : ℂ) * ξ v * | |
| conj (γ ((n : ℤ) + v))) = 0 | |
| calc | |
| _ = ∑' v : ℤ, (ξ v / (Fx v : ℂ)) * | |
| ((Fy ((n : ℤ) + v) : ℂ) * conj (γ ((n : ℤ) + v))) := by | |
| apply tsum_congr | |
| intro v | |
| exact (InvariantProjection.convolution_coefficient_factorization Fx Fy | |
| (fun v => ξ v) (fun v => γ v) (n : ℤ) v).symm | |
| _ = 0 := hzero) | |
| (by | |
| intro v hv | |
| have hvw : ¬ (|v| ≤ (K : ℤ) ∧ |(n : ℤ) + v| ≤ (K : ℤ)) := by | |
| intro hw | |
| apply hv | |
| simp only [window, Finset.mem_filter, Finset.mem_Icc] | |
| exact ⟨abs_le.mp hw.1, hw.2⟩ | |
| have hk := InvariantProjection.omitted_index_selector (K : ℤ) (n : ℤ) v hvw | |
| have hb := (InvariantProjection.omitted_index_ratio_bound (K : ℤ) (n : ℤ) v | |
| (fun k => Fy k / Fx k) _ (hratio v).2.1).trans (htail _ hk) | |
| simpa only [selector, hmask] using hb) | |
| simp only [hmask] at hdom | |
| rw [tsum_omittedEvent H₁ K n ω | |
| (fun v => ‖ξ v‖ * ‖γ ((Equiv.addLeft (n : ℤ)) v)‖), | |
| tsum_omittedEvent H₂ K n ω | |
| (fun v => ‖ξ v‖ * ‖γ ((Equiv.addLeft (n : ℤ)) v)‖)] at hdom | |
| convert hdom using 1 | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro v hv | |
| change (ξ v / (Fx v : ℂ)) * | |
| ((Fy (-(-(n : ℤ) - v)) : ℂ) * conj (γ (-(-(n : ℤ) - v)))) = _ | |
| have hsign : -(-(n : ℤ) - v) = (n : ℤ) + v := by omega | |
| rw [hsign] | |
| exact InvariantProjection.convolution_coefficient_factorization Fx Fy (fun v => ξ v) (fun v => γ v) (n : ℤ) v | |
| all_goals rfl | |
| end InvariantProjection.ConditionalConvolution | |
| end OAI | |