File size: 3,014 Bytes
f1f7953 | 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 | import OAI.Analysis.IrrationalRotation.Model.CircleParameters
import OAI.Analysis.IrrationalRotation.Model.ScalarSampling
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
import Mathlib.Tactic
namespace OAI
open MeasureTheory
namespace Invariant233.CircleParameterData
theorem kernel_eq_manuscriptKernel (m : ℕ) (z v : ℝ)
(hv : v ∈ Set.Ico (0 : ℝ) 1) :
kernel m z v = ScalarSampling.manuscriptKernel m z v := by
have he : Int.fract v = v := Int.fract_eq_self.mpr hv
simp only [kernel, kernelWidth, delta, ScalarSampling.manuscriptKernel,
ScalarSampling.circleKernel, ScalarSampling.rawKernel, ScalarSampling.delta, he,
Set.indicator_apply, Set.mem_Ioo, one_div]
split_ifs <;> simp_all
theorem starting_kernel_fract (m : ℕ) (z v : ℝ) :
orientedKernel false m z (Int.fract v) =
ScalarSampling.manuscriptKernel m z (Int.fract v) := by
simpa only [orientedKernel, Bool.false_eq_true, ↓reduceIte] using
kernel_eq_manuscriptKernel m z (Int.fract v)
⟨Int.fract_nonneg v, Int.fract_lt_one v⟩
theorem endpoint_kernel_fract (m : ℕ) (z v : ℝ) :
orientedKernel true m z (Int.fract v) =
ScalarSampling.manuscriptEndpointKernel m z (Int.fract v) := by
simpa only [orientedKernel, ↓reduceIte, ScalarSampling.manuscriptEndpointKernel] using
kernel_eq_manuscriptKernel m (z - Int.fract v) (Int.fract v)
⟨Int.fract_nonneg v, Int.fract_lt_one v⟩
theorem measurable_rep : Measurable rep := by
exact measurable_subtype_coe.comp (AddCircle.measurableEquivIco (T := (1 : ℝ)) 0).measurable
theorem measurable_coordinate (p : CircleParameterData) (i : ℕ) :
Measurable (p.coordinate i) := by
exact measurable_rep.comp (continuous_nsmul (p.q i)).measurable
theorem measurable_kernel (m : ℕ) (v : ℝ) : Measurable (fun z => kernel m z v) := by
have hd : Measurable delta := measurable_fract.min (measurable_const.sub measurable_fract)
unfold kernel
apply Measurable.ite
· exact (measurableSet_le measurable_const hd).inter
((MeasurableSet.const (0 < v)).inter
(measurableSet_lt measurable_const (measurable_const.mul hd)))
· exact measurable_const.div (measurable_const.mul hd)
· exact measurable_const
theorem measurable_orientedKernel (endpoint : Bool) (m : ℕ) (v : ℝ) :
Measurable (fun z => orientedKernel endpoint m z v) := by
cases endpoint
· exact measurable_kernel m v
· exact (measurable_kernel m v).comp (measurable_id.sub_const v)
theorem measurable_samplingAverage (p : CircleParameterData) (θ : ℝ) (j r : ℕ)
(endpoint : Bool) : Measurable (p.samplingAverage θ j r endpoint) := by
unfold samplingAverage
apply Measurable.const_mul
apply Finset.measurable_sum
intro n hn
apply Finset.measurable_prod
intro i hi
exact (measurable_orientedKernel endpoint (p.m j)
(Int.fract ((n : ℝ) * p.alpha θ i))).comp (p.measurable_coordinate i)
end Invariant233.CircleParameterData
end OAI
|