openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleSamplingKernelBridge.lean
| 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 | |