File size: 3,611 Bytes
67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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