openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterSamplingTests.lean
| import OAI.Analysis.IrrationalRotation.Model.CharacterSampling | |
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingLimit | |
| import OAI.Analysis.IrrationalRotation.Sampling.ScalarSamplingContinuity | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleRepresentative | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped BigOperators Topology | |
| namespace Invariant233.CharacterSampling | |
| lemma fract_sub_fract (z v : ℝ) : Int.fract (z - Int.fract v) = Int.fract (z - v) := by | |
| have heq : z - Int.fract v = z - v + (⌊v⌋ : ℤ) := by | |
| rw [Int.fract] | |
| ring | |
| rw [heq, Int.fract_add_intCast] | |
| theorem kernel_fract_right (endpoint : Bool) (m : ℕ) (z v : ℝ) : | |
| kernel endpoint m z (Int.fract v) = kernel endpoint m z v := by | |
| cases endpoint <;> | |
| simp only [kernel, Bool.false_eq_true, ↓reduceIte, ScalarSampling.manuscriptEndpointKernel, | |
| ScalarSampling.manuscriptKernel, ScalarSampling.circleKernel, Int.fract_fract, | |
| fract_sub_fract] | |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := | |
| ⟨by simp [AddCircle.measure_univ]⟩ | |
| lemma measurable_representative : Measurable representative := | |
| measurable_subtype_coe.comp (AddCircle.measurableEquivIco (1 : ℝ) 0).measurable | |
| lemma continuousAt_representative {x : UnitAddCircle} (hx : x ≠ 0) : | |
| ContinuousAt representative x := | |
| continuousAt_subtype_val.comp (AddCircle.continuousAt_equivIco (1 : ℝ) 0 hx) | |
| lemma measurable_kernel (endpoint : Bool) (m : ℕ) : | |
| Measurable (fun p : ℝ × ℝ => kernel endpoint m p.1 p.2) := by | |
| cases endpoint | |
| · exact ScalarSampling.measurable_circleKernel _ _ | |
| · exact (ScalarSampling.measurable_circleKernel _ _).comp | |
| ((measurable_fst.sub measurable_snd).prodMk measurable_snd) | |
| lemma kernel_nonneg (endpoint : Bool) {m : ℕ} (hm : 200 ≤ m) (z v : ℝ) : | |
| 0 ≤ kernel endpoint m z v := by | |
| have hh := (ScalarSampling.manuscript_parameters hm).1.le | |
| cases endpoint <;> exact ScalarSampling.rawKernel_nonneg hh | |
| ⟨Int.fract_nonneg _, (Int.fract_lt_one _).le⟩ | |
| lemma kernel_le_height (endpoint : Bool) {m : ℕ} (hm : 200 ≤ m) (z v : ℝ) : | |
| kernel endpoint m z v ≤ (m : ℝ) ^ (13 / 5 : ℝ) := by | |
| cases endpoint <;> exact ScalarSampling.manuscriptKernel_le_height hm _ _ | |
| lemma kernel_abs_le_height (endpoint : Bool) {m : ℕ} (hm : 200 ≤ m) (z v : ℝ) : | |
| |kernel endpoint m z v| ≤ (m : ℝ) ^ (13 / 5 : ℝ) := by | |
| rw [abs_of_nonneg (kernel_nonneg endpoint hm z v)] | |
| exact kernel_le_height endpoint hm z v | |
| theorem ae_joint_continuousAt_kernel {m : ℕ} (hm : 200 ≤ m) | |
| (endpoint : Bool) (n τ : ℝ) (hphase : Int.fract (n * τ) ≠ 0) : | |
| ∀ᵐ x : UnitAddCircle, | |
| ContinuousAt (fun p : ℝ × UnitAddCircle => | |
| kernel endpoint m (representative p.2) (n * p.1)) (τ, x) := by | |
| have hreal : ∀ᵐ z : ℝ ∂volume, | |
| ContinuousAt (fun p : ℝ × ℝ => kernel endpoint m p.1 (n * p.2)) (z, τ) := by | |
| cases endpoint | |
| · exact ScalarSampling.ae_joint_continuousAt_manuscriptPower hm n τ hphase | |
| · exact ScalarSampling.ae_joint_continuousAt_manuscriptEndpointPower hm n τ hphase | |
| have hpull := CocycleSupport.measurePreserving_circleRepresentative.quasiMeasurePreserving.ae | |
| (ae_restrict_of_ae (s := Ico (0 : ℝ) 1) hreal) | |
| have hnotzero := CocycleSupport.measurePreserving_circleRepresentative.quasiMeasurePreserving.ae | |
| (ae_restrict_of_ae (s := Ico (0 : ℝ) 1) ((volume : Measure ℝ).ae_ne 0)) | |
| filter_upwards [hpull, hnotzero] with x hx hrepzero | |
| have hzero : x ≠ 0 := by | |
| intro heq | |
| subst x | |
| apply hrepzero | |
| exact AddCircle.equivIco_coe_of_mem (p := (1 : ℝ)) (a := 0) (by norm_num) | |
| have hc : ContinuousAt | |
| (fun p : ℝ × UnitAddCircle => (representative p.2, p.1)) (τ, x) := | |
| ((continuousAt_representative hzero).comp continuous_snd.continuousAt).prodMk | |
| continuous_fst.continuousAt | |
| exact hx.comp_of_eq hc rfl | |
| lemma measurable_squaredError (m : ℕ) (powers : Finset ℕ) (endpoint : Bool) : | |
| Measurable (squaredError m powers endpoint) := by | |
| apply Measurable.pow_const | |
| apply Measurable.sub _ measurable_const | |
| apply measurable_const.mul | |
| apply Finset.measurable_sum | |
| intro n hn | |
| apply Finset.measurable_prod | |
| intro i hi | |
| exact (measurable_kernel endpoint m).comp | |
| ((measurable_representative.comp ((measurable_pi_apply i).comp measurable_snd)).prodMk | |
| (measurable_const.mul ((measurable_pi_apply i).comp measurable_fst))) | |
| theorem ae_continuousAt_coordinateFactor {m : ℕ} (hm : 200 ≤ m) | |
| (endpoint : Bool) (n : ℕ) (phase : Fin m → ℝ) (i : Fin m) | |
| (hphase : Int.fract ((n : ℝ) * phase i) ≠ 0) : | |
| ∀ᵐ y : UnitAddTorus (Fin m), ContinuousAt | |
| (fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) => | |
| kernel endpoint m (representative (p.2 i)) ((n : ℝ) * p.1 i)) | |
| (phase, y) := by | |
| have heval : ∀ᵐ y : UnitAddTorus (Fin m), ContinuousAt | |
| (fun p : ℝ × UnitAddCircle => kernel endpoint m (representative p.2) ((n : ℝ) * p.1)) | |
| (phase i, y i) := | |
| (measurePreserving_eval (fun _ : Fin m => (volume : Measure UnitAddCircle)) i).quasiMeasurePreserving.ae | |
| (ae_joint_continuousAt_kernel hm endpoint (n : ℝ) (phase i) hphase) | |
| filter_upwards [heval] with y hy | |
| have hc : ContinuousAt | |
| (fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) => (p.1 i, p.2 i)) (phase,y) := | |
| ((continuous_apply i).continuousAt.comp continuous_fst.continuousAt).prodMk | |
| ((continuous_apply i).continuousAt.comp continuous_snd.continuousAt) | |
| exact ContinuousAt.comp (x := (phase,y)) hy hc | |
| theorem ae_continuousAt_squaredError {m : ℕ} (hm : 200 ≤ m) | |
| (powers : Finset ℕ) (endpoint : Bool) (phase : Fin m → ℝ) | |
| (hphase : ∀ n ∈ powers, ∀ i, Int.fract ((n : ℝ) * phase i) ≠ 0) : | |
| ∀ᵐ y : UnitAddTorus (Fin m), | |
| ContinuousAt (squaredError m powers endpoint) (phase, y) := by | |
| have hall : ∀ᵐ y : UnitAddTorus (Fin m), ∀ n ∈ powers, ∀ i : Fin m, | |
| ContinuousAt | |
| (fun p : (Fin m → ℝ) × UnitAddTorus (Fin m) => | |
| kernel endpoint m (representative (p.2 i)) ((n : ℝ) * p.1 i)) | |
| (phase, y) := by | |
| rw [Finset.eventually_all] | |
| intro n hn | |
| exact Filter.eventually_all.mpr (fun i => ae_continuousAt_coordinateFactor hm endpoint n phase i (hphase n hn i)) | |
| filter_upwards [hall] with y hy | |
| apply ContinuousAt.pow | |
| apply ContinuousAt.sub _ continuousAt_const | |
| apply ContinuousAt.const_mul | |
| exact tendsto_finsetSum powers (fun n hn => | |
| tendsto_finsetProd Finset.univ (fun i _ => hy n hn i)) | |
| lemma average_nonneg {m : ℕ} (hm : 200 ≤ m) (powers : Finset ℕ) | |
| (endpoint : Bool) (phase : Fin m → ℝ) (spatial : UnitAddTorus (Fin m)) : | |
| 0 ≤ average m powers endpoint phase spatial := by | |
| apply mul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _)) | |
| apply Finset.sum_nonneg | |
| intro n hn | |
| exact Finset.prod_nonneg (fun i _ => kernel_nonneg endpoint hm _ _) | |
| lemma average_le_height {m : ℕ} (hm : 200 ≤ m) (powers : Finset ℕ) | |
| (endpoint : Bool) (phase : Fin m → ℝ) (spatial : UnitAddTorus (Fin m)) : | |
| average m powers endpoint phase spatial ≤ ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m := by | |
| by_cases hp : powers.Nonempty | |
| · have hcard : (0 : ℝ) < powers.card := by exact_mod_cast hp.card_pos | |
| have hs : (∑ n ∈ powers, ∏ i : Fin m, | |
| kernel endpoint m (representative (spatial i)) ((n : ℝ) * phase i)) ≤ | |
| (powers.card : ℝ) * ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m := by | |
| calc | |
| _ ≤ ∑ _n ∈ powers, ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m := by | |
| apply Finset.sum_le_sum | |
| intro n hn | |
| calc | |
| _ ≤ ∏ _i : Fin m, (m : ℝ) ^ (13 / 5 : ℝ) := | |
| Finset.prod_le_prod₀ (fun i _ => kernel_nonneg endpoint hm _ _) | |
| (fun i _ => kernel_le_height endpoint hm _ _) | |
| _ = _ := by simp | |
| _ = _ := by simp | |
| calc | |
| average m powers endpoint phase spatial ≤ | |
| (powers.card : ℝ)⁻¹ * ((powers.card : ℝ) * ((m : ℝ) ^ (13 / 5 : ℝ)) ^ m) := | |
| mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr hcard.le) | |
| _ = _ := by rw [← mul_assoc, inv_mul_cancel₀ hcard.ne', one_mul] | |
| · have hempty : powers = ∅ := Finset.not_nonempty_iff_eq_empty.mp hp | |
| simp only [hempty, average, Finset.card_empty, Nat.cast_zero, inv_zero, Finset.sum_empty, mul_zero] | |
| positivity | |
| theorem squaredError_abs_le {m : ℕ} (hm : 200 ≤ m) (powers : Finset ℕ) | |
| (endpoint : Bool) (p : (Fin m → ℝ) × UnitAddTorus (Fin m)) : | |
| |squaredError m powers endpoint p| ≤ (((m : ℝ) ^ (13 / 5 : ℝ)) ^ m + 1) ^ 2 := by | |
| rw [squaredError, abs_pow] | |
| apply pow_le_pow_left₀ (abs_nonneg _) | |
| calc | |
| |average m powers endpoint p.1 p.2 - 1| ≤ | |
| |average m powers endpoint p.1 p.2| + |(1 : ℝ)| := by simpa using (abs_sub_le (average m powers endpoint p.1 p.2) 0 1) | |
| _ = average m powers endpoint p.1 p.2 + 1 := by | |
| rw [abs_of_nonneg (average_nonneg hm powers endpoint p.1 p.2), abs_one] | |
| _ ≤ _ := by linarith [average_le_height hm powers endpoint p.1 p.2] | |
| theorem tendsto_integral_squaredError {m : ℕ} (hm : 200 ≤ m) | |
| (powers : Finset ℕ) (endpoint : Bool) | |
| (q : ℕ → ℕ → ℕ) (hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j)) | |
| (hgap : ∀ i, i + 1 < m → Tendsto | |
| (fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop) | |
| {phase : ℕ → Fin m → ℝ} {target : Fin m → ℝ} | |
| (hphase : Tendsto phase atTop (𝓝 target)) | |
| (hcut : ∀ n ∈ powers, ∀ i, Int.fract ((n : ℝ) * target i) ≠ 0) : | |
| Tendsto (fun j => ∫ x : UnitAddCircle, | |
| squaredError m powers endpoint | |
| (phase j, fun i : Fin m => (q j i.val : ℤ) • x)) atTop | |
| (𝓝 (∫ y : UnitAddTorus (Fin m), squaredError m powers endpoint (target, y))) := by | |
| exact CharacterLaws.tendsto_integral_character_phase_of_ae_continuous q m hq hmono hgap | |
| hphase (measurable_squaredError m powers endpoint) | |
| (ae_continuousAt_squaredError hm powers endpoint target hcut) | |
| (sq_nonneg _) (squaredError_abs_le hm powers endpoint) | |
| theorem integrable_squaredError_comp {X : Type*} [MeasurableSpace X] | |
| {μ : Measure X} [IsFiniteMeasure μ] {m : ℕ} (hm : 200 ≤ m) | |
| (powers : Finset ℕ) (endpoint : Bool) | |
| {F : X → (Fin m → ℝ) × UnitAddTorus (Fin m)} (hF : Measurable F) : | |
| Integrable (fun x => squaredError m powers endpoint (F x)) μ := by | |
| apply Integrable.of_bound ((measurable_squaredError m powers endpoint).comp hF).aestronglyMeasurable | |
| ((((m : ℝ) ^ (13 / 5 : ℝ)) ^ m + 1) ^ 2) | |
| exact Eventually.of_forall (fun x => by | |
| simpa only [Real.norm_eq_abs, Function.comp_apply] using squaredError_abs_le hm powers endpoint (F x)) | |
| theorem lintegral_squaredError_eq {X : Type*} [MeasurableSpace X] | |
| {μ : Measure X} [IsFiniteMeasure μ] {m : ℕ} (hm : 200 ≤ m) | |
| (powers : Finset ℕ) (endpoint : Bool) | |
| {F : X → (Fin m → ℝ) × UnitAddTorus (Fin m)} (hF : Measurable F) : | |
| (∫⁻ x, ENNReal.ofReal (squaredError m powers endpoint (F x)) ∂μ) = | |
| ENNReal.ofReal (∫ x, squaredError m powers endpoint (F x) ∂μ) := by | |
| apply (ofReal_integral_eq_lintegral_ofReal | |
| (integrable_squaredError_comp hm powers endpoint hF) _).symm | |
| exact Eventually.of_forall (fun x => sq_nonneg _) | |
| end Invariant233.CharacterSampling | |
| end | |
| end OAI | |