openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterSamplingTrial.lean
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingTests | |
| import OAI.Analysis.IrrationalRotation.Model.TrialSampling | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped BigOperators | |
| namespace Invariant233.CharacterSampling | |
| theorem kernel_eq_parameter_orientedKernel (endpoint : Bool) (m : ℕ) (z v : ℝ) : | |
| kernel endpoint m z v = | |
| CircleParameterData.orientedKernel endpoint m z (Int.fract v) := by | |
| cases endpoint <;> | |
| simp only [kernel, CircleParameterData.orientedKernel, Bool.false_eq_true, | |
| ↓reduceIte, ScalarSampling.manuscriptEndpointKernel, | |
| ScalarSampling.manuscriptKernel, ScalarSampling.circleKernel, | |
| ScalarSampling.rawKernel, ScalarSampling.delta, CircleParameterData.kernel, | |
| CircleParameterData.delta, CircleParameterData.kernelWidth, | |
| Set.indicator_apply, Set.mem_Ioo, one_div, fract_sub_fract, ite_and] | |
| theorem average_eq_trialAverage (m : ℕ) (samples : Finset ℕ) (endpoint : Bool) | |
| (phase : Fin m → ℝ) (z : UnitAddTorus (Fin m)) : | |
| average m samples endpoint phase z = TrialSampling.average m samples endpoint phase z := by | |
| unfold average TrialSampling.average | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| apply Finset.prod_congr rfl | |
| intro i hi | |
| exact kernel_eq_parameter_orientedKernel endpoint m _ _ | |
| theorem trialSpatialLaw_eq_volume (m : ℕ) : | |
| TrialSampling.spatialLaw m = (volume : Measure (UnitAddTorus (Fin m))) := rfl | |
| theorem trialSquaredError_eq_ofReal_integral {m : ℕ} (hm : 200 ≤ m) | |
| (samples : Finset ℕ) (endpoint : Bool) (phase : Fin m → ℝ) : | |
| TrialSampling.squaredError m samples endpoint phase = | |
| ENNReal.ofReal (∫ z : UnitAddTorus (Fin m), | |
| squaredError m samples endpoint (phase, z)) := by | |
| rw [TrialSampling.squaredError, trialSpatialLaw_eq_volume] | |
| simp_rw [← average_eq_trialAverage] | |
| exact lintegral_squaredError_eq hm samples endpoint | |
| (F := fun z : UnitAddTorus (Fin m) => (phase,z)) | |
| (measurable_const.prodMk measurable_id) | |
| theorem trialSquaredError_lt_iff {m : ℕ} (hm : 200 ≤ m) | |
| (samples : Finset ℕ) (endpoint : Bool) (phase : Fin m → ℝ) | |
| {δ : ℝ} (hδ : 0 < δ) : | |
| TrialSampling.squaredError m samples endpoint phase < ENNReal.ofReal δ ↔ | |
| (∫ z : UnitAddTorus (Fin m), | |
| squaredError m samples endpoint (phase, z)) < δ := by | |
| rw [trialSquaredError_eq_ofReal_integral hm, ENNReal.ofReal_lt_ofReal_iff hδ] | |
| end Invariant233.CharacterSampling | |
| end | |
| end OAI | |