| import OAI.Analysis.FiniteFactor.OrbitSampling.TestedMarginals |
| import OAI.Analysis.FiniteFactor.OrbitSampling.FirstMarginal |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| / |
| coordinate. Whole-circle tests fill the remaining coordinates. -/ |
| abbrev SourceSamplingRationalTest := |
| Σ K : Finset SourceCoordinate, K → (ℚ × ℚ) × (ℚ × ℚ) |
| |
| instance : Countable SourceSamplingRationalTest := inferInstance |
| |
| def sourceSamplingRationalEndpoints (T : SourceSamplingRationalTest) (i : SourceCoordinate) : |
| (ℚ × ℚ) × (ℚ × ℚ) := |
| if hi : i ∈ T.1 then T.2 ⟨i,hi⟩ else ((0,1),(0,1)) |
| |
| def sourceSamplingRationalX (T : SourceSamplingRationalTest) (i : SourceCoordinate) : Set ℝ := |
| Set.Icc (sourceSamplingRationalEndpoints T i).1.1 (sourceSamplingRationalEndpoints T i).1.2 |
| |
| def sourceSamplingRationalY (T : SourceSamplingRationalTest) (i : SourceCoordinate) : Set ℝ := |
| Set.Icc (sourceSamplingRationalEndpoints T i).2.1 (sourceSamplingRationalEndpoints T i).2.2 |
| |
| theorem sourceSamplingRationalTests_outside (T : SourceSamplingRationalTest) (stage : ℕ) |
| (hK : T.1 ⊆ sourceSamplingPrefix stage) : |
| ∀ i ∉ sourceSamplingPrefix stage, |
| Set.Ico (0 : ℝ) 1 ⊆ sourceSamplingRationalX T i ∧ |
| Set.Ico (0 : ℝ) 1 ⊆ sourceSamplingRationalY T i := by |
| intro i hi |
| have hnot : i ∉ T.1 := fun h => hi (hK h) |
| simp only [sourceSamplingRationalX,sourceSamplingRationalY,sourceSamplingRationalEndpoints, |
| dite_eq_right hnot,Rat.cast_zero,Rat.cast_one] |
| exact ⟨Set.Ico_subset_Icc_self,Set.Ico_subset_Icc_self⟩ |
| |
| theorem sourceSamplingPrefix_eventually_covers (K : Finset SourceCoordinate) : |
| ∀ᶠ stage : ℕ in atTop, K ⊆ sourceSamplingPrefix stage := by |
| filter_upwards [eventually_ge_atTop (K.sup (fun i => i.1))] with stage hstage |
| intro i hi |
| apply (CouplingParameters.mem_blockPrefix sourceBlockSize (stage+1) i).mpr |
| exact Nat.lt_succ_of_le ((Finset.le_sup (f := fun i => i.1) hi).trans hstage) |
| |
| / |
| all rational cylinder tests. It includes the actual tested measure identity |
| at every admissible stage, as well as the full-error convergence and L2 bound. -/ |
| theorem sourceSampling_rational_event : |
| ∀ᵐ θ ∂sourceAngleLaw, ∀ (I : Finset SourceCoordinate) (r : Bool) |
| (T : SourceSamplingRationalTest), |
| Tendsto (fun stage => sourceSamplingSquaredError stage r (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) θ) atTop (𝓝 0) ∧ |
| ∃ C : ℝ, ∀ stage, |
| MemLp (sourceSamplingEmpiricalDensity stage r (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) θ) 2 sourceSamplingHaar ∧ |
| (∫ x, sourceSamplingEmpiricalDensity stage r (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) θ x ^ 2 ∂sourceSamplingHaar) ≤ C ∧ |
| (T.1 ⊆ sourceSamplingPrefix stage → |
| ((sourceSamplingUntrimmed I stage θ).restrict |
| (sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).map |
| (sourceSamplingEndpoint r) = |
| sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal |
| (sourceSamplingEmpiricalDensity stage r (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) θ x))) := by |
| apply ae_all_iff.mpr |
| intro I |
| apply ae_all_iff.mpr |
| intro r |
| apply ae_all_iff.mpr |
| intro T |
| filter_upwards [sourceSamplingSquaredError_ae r (fun i => decide (i ∈ I)) |
| (fun i => (sourceSamplingRationalEndpoints T i).1.1) |
| (fun i => (sourceSamplingRationalEndpoints T i).1.2) |
| (fun i => (sourceSamplingRationalEndpoints T i).2.1) |
| (fun i => (sourceSamplingRationalEndpoints T i).2.2), |
| sourceSamplingEmpiricalDensity_uniform r (fun i => decide (i ∈ I)) |
| (fun i => (sourceSamplingRationalEndpoints T i).1.1) |
| (fun i => (sourceSamplingRationalEndpoints T i).1.2) |
| (fun i => (sourceSamplingRationalEndpoints T i).2.1) |
| (fun i => (sourceSamplingRationalEndpoints T i).2.2)] with θ he hu |
| obtain ⟨C,hC⟩ := hu |
| refine ⟨he.2,C,fun stage => ⟨(hC stage).1,(hC stage).2,?_⟩⟩ |
| intro hK |
| exact sourceSamplingUntrimmed_tested_marginal I stage r _ _ |
| (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc) |
| (sourceSamplingRationalTests_outside T stage hK) θ |
| |
| / |
| conull event, simultaneously for all finite I. -/ |
| theorem sourceSampling_both_marginals_event : |
| ∀ᵐ θ ∂sourceAngleLaw, ∀ (I : Finset SourceCoordinate) (r : Bool), |
| ∃ C : ℝ, ∀ stage, ∃ f : (SourceCoordinate → ℝ) → ℝ, |
| (sourceSamplingUntrimmed I stage θ).map (sourceSamplingEndpoint r) = |
| sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal (f x)) ∧ |
| MemLp f 2 sourceSamplingHaar ∧ (∫ x, f x^2 ∂sourceSamplingHaar) ≤ C := by |
| apply ae_all_iff.mpr |
| intro I |
| apply ae_all_iff.mpr |
| intro r |
| cases r |
| · exact sourceSamplingUntrimmed_first_L2 I |
| · exact sourceSamplingUntrimmed_second_L2 I |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
