| import OAI.Analysis.FiniteFactor.OrbitSampling.Cube | |
| import Mathlib.Topology.Instances.Rat | |
| namespace OAI | |
| namespace FiniteFactor | |
| open MeasureTheory Filter Finset | |
| open scoped Classical ENNReal Topology | |
| noncomputable section | |
| /-- Finite cylinders in the full real product: coordinates outside the | |
| finite support are unrestricted. On the conull cube these are the original | |
| rational tests. The ambient real product is not asserted to be compact. -/ | |
| def sourceSamplingCylinder (T : SourceSamplingRationalTest) : Set SourceSamplingPair := | |
| {p | ∀ i ∈ T.1, p.1 i ∈ sourceSamplingRationalX T i ∧ p.2 i ∈ sourceSamplingRationalY T i} | |
| theorem sourceSamplingCylinder_measurable (T : SourceSamplingRationalTest) : | |
| MeasurableSet (sourceSamplingCylinder T) := by | |
| simp only [sourceSamplingCylinder,Set.ofPred_forall] | |
| exact MeasurableSet.iInter (fun i => MeasurableSet.iInter (fun _ => | |
| (measurableSet_Icc.preimage ((measurable_pi_apply i).comp measurable_fst)).inter | |
| (measurableSet_Icc.preimage ((measurable_pi_apply i).comp measurable_snd)))) | |
| theorem sourceSamplingCylinder_on_cube (T : SourceSamplingRationalTest) (p : SourceSamplingPair) | |
| (hp : p ∈ sourceSamplingCube) : | |
| p ∈ sourceSamplingCylinder T ↔ | |
| p ∈ sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T) := by | |
| constructor | |
| · intro h i | |
| by_cases hi : i ∈ T.1 | |
| · exact h i hi | |
| · simpa [sourceSamplingRationalX,sourceSamplingRationalY,sourceSamplingRationalEndpoints,hi] | |
| using hp i | |
| · intro h i _ | |
| exact h i | |
| def sourceSamplingTestMeet (T U : SourceSamplingRationalTest) : SourceSamplingRationalTest := | |
| ⟨T.1 ∪ U.1,fun i => | |
| if i.1 ∈ T.1 then | |
| if i.1 ∈ U.1 then | |
| ((max (sourceSamplingRationalEndpoints T i).1.1 (sourceSamplingRationalEndpoints U i).1.1, | |
| min (sourceSamplingRationalEndpoints T i).1.2 (sourceSamplingRationalEndpoints U i).1.2), | |
| (max (sourceSamplingRationalEndpoints T i).2.1 (sourceSamplingRationalEndpoints U i).2.1, | |
| min (sourceSamplingRationalEndpoints T i).2.2 (sourceSamplingRationalEndpoints U i).2.2)) | |
| else sourceSamplingRationalEndpoints T i | |
| else sourceSamplingRationalEndpoints U i⟩ | |
| theorem sourceSamplingCylinder_inter (T U : SourceSamplingRationalTest) : | |
| sourceSamplingCylinder (sourceSamplingTestMeet T U) = | |
| sourceSamplingCylinder T ∩ sourceSamplingCylinder U := by | |
| have hcoord (p : SourceSamplingPair) (i : SourceCoordinate) : | |
| (i ∈ T.1 ∪ U.1 → p.1 i ∈ sourceSamplingRationalX (sourceSamplingTestMeet T U) i ∧ | |
| p.2 i ∈ sourceSamplingRationalY (sourceSamplingTestMeet T U) i) ↔ | |
| (i ∈ T.1 → p.1 i ∈ sourceSamplingRationalX T i ∧ p.2 i ∈ sourceSamplingRationalY T i) ∧ | |
| (i ∈ U.1 → p.1 i ∈ sourceSamplingRationalX U i ∧ p.2 i ∈ sourceSamplingRationalY U i) := by | |
| by_cases ht : i ∈ T.1 <;> by_cases hu : i ∈ U.1 <;> | |
| simp [sourceSamplingRationalX,sourceSamplingRationalY,sourceSamplingRationalEndpoints, | |
| sourceSamplingTestMeet,ht,hu,Set.mem_Icc] | |
| tauto | |
| ext p | |
| change (∀ i, i ∈ T.1 ∪ U.1 → _) ↔ (∀ i, i ∈ T.1 → _) ∧ (∀ i, i ∈ U.1 → _) | |
| constructor | |
| · intro h | |
| exact ⟨fun i => ((hcoord p i).mp (h i)).1,fun i => ((hcoord p i).mp (h i)).2⟩ | |
| · rintro ⟨hT,hU⟩ i | |
| exact (hcoord p i).mpr ⟨hT i,hU i⟩ | |
| theorem sourceSamplingCylinder_piSystem : IsPiSystem (Set.range sourceSamplingCylinder) := by | |
| rintro _ ⟨T,rfl⟩ _ ⟨U,rfl⟩ _ | |
| exact ⟨sourceSamplingTestMeet T U,sourceSamplingCylinder_inter T U⟩ | |
| end | |
| end FiniteFactor | |
| end OAI | |