openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterLawConvergence.lean
| import OAI.Analysis.IrrationalRotation.Model.CharacterLaw | |
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterWeakConvergence | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| namespace CharacterLaws | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| theorem tendsto_characterLaw | |
| (q : ℕ → ℕ → ℕ) (m : 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) : | |
| Tendsto (fun j => characterLaw (fun i : Fin m => q j i.val)) atTop | |
| (𝓝 (torusHaar m)) := by | |
| apply (ProbabilityMeasure.tendsto_iff_forall_integral_rclike_tendsto ℂ).mpr | |
| intro f | |
| have h := CharacterWeakConvergence.tendsto_integral_joint_characters | |
| q m hq hmono hgap f.toContinuousMap | |
| have hmap (j : ℕ) : | |
| (∫ y, f y ∂(characterLaw (fun i : Fin m => q j i.val) : Measure _)) = | |
| ∫ x : AddCircle (1 : ℝ), f (fun i : Fin m => (q j i.val : ℤ) • x) := by | |
| change (∫ y, f y ∂Measure.map (characterMap (fun i : Fin m => q j i.val)) volume) = _ | |
| rw [integral_map | |
| (characterMap (fun i : Fin m => q j i.val)).continuous.measurable.aemeasurable | |
| f.continuous.aestronglyMeasurable] | |
| rfl | |
| simpa only [hmap, BoundedContinuousFunction.coe_toContinuousMap, | |
| torusHaar, ProbabilityMeasure.coe_mk] using h | |
| theorem tendsto_apply_of_uniform_bound | |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] | |
| (L : ℕ → E →L[ℂ] ℂ) (M : E →L[ℂ] ℂ) | |
| (hL : ∀ j, ‖L j‖ ≤ 1) | |
| (hconv : ∀ f, Tendsto (fun j => L j f) atTop (𝓝 (M f))) | |
| {f : ℕ → E} {g : E} (hf : Tendsto f atTop (𝓝 g)) : | |
| Tendsto (fun j => L j (f j)) atTop (𝓝 (M g)) := by | |
| have hnorm : Tendsto (fun j => ‖f j - g‖) atTop (𝓝 0) := by | |
| have hg : Tendsto (fun _ : ℕ => g) atTop (𝓝 g) := tendsto_const_nhds | |
| simpa using (hf.sub hg).norm | |
| have hsmall : Tendsto (fun j => L j (f j - g)) atTop (𝓝 0) := by | |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr | |
| apply squeeze_zero (fun _ => norm_nonneg _) (fun j => ?_) hnorm | |
| calc | |
| ‖L j (f j - g)‖ ≤ ‖L j‖ * ‖f j - g‖ := (L j).le_opNorm _ | |
| _ ≤ 1 * ‖f j - g‖ := mul_le_mul_of_nonneg_right (hL j) (norm_nonneg _) | |
| _ = ‖f j - g‖ := one_mul _ | |
| simpa only [map_sub, sub_add_cancel, zero_add] using hsmall.add (hconv g) | |
| theorem tendsto_integral_character_phase | |
| {Z : Type*} [TopologicalSpace Z] | |
| (q : ℕ → ℕ → ℕ) (m : 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) | |
| {a : ℕ → Z} {b : Z} (ha : Tendsto a atTop (𝓝 b)) | |
| (f : C(Z × UnitAddTorus (Fin m), ℂ)) : | |
| Tendsto (fun j => ∫ x : AddCircle (1 : ℝ), | |
| f (a j, fun i : Fin m => (q j i.val : ℤ) • x)) atTop | |
| (𝓝 (∫ y : UnitAddTorus (Fin m), f (b, y))) := by | |
| let : IsProbabilityMeasure (volume : Measure (AddCircle (1 : ℝ))) := | |
| circleHaar.property | |
| let L : ℕ → C(UnitAddTorus (Fin m), ℂ) →L[ℂ] ℂ := | |
| fun j => CharacterWeakConvergence.pullbackIntegral volume | |
| (characterMap (fun i : Fin m => q j i.val)) | |
| let M : C(UnitAddTorus (Fin m), ℂ) →L[ℂ] ℂ := | |
| CharacterWeakConvergence.pullbackIntegral volume (ContinuousMap.id _) | |
| exact tendsto_apply_of_uniform_bound L M | |
| (fun j => CharacterWeakConvergence.pullbackIntegral_norm_le volume _) | |
| (fun g => CharacterWeakConvergence.tendsto_integral_joint_characters | |
| q m hq hmono hgap g) | |
| (f.curry.continuous.tendsto b |>.comp ha) | |
| theorem integral_characterPhaseLaw {Z : Type*} [TopologicalSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] {m : ℕ} (q : Fin m → ℕ) (a : Z) | |
| (f : C(Z × UnitAddTorus (Fin m), ℂ)) : | |
| (∫ z, f z ∂(characterPhaseLaw q a : Measure _)) = | |
| ∫ x : AddCircle (1 : ℝ), f (a, characterMap q x) := by | |
| change (∫ z, f z ∂Measure.map (fun x => (a, characterMap q x)) volume) = _ | |
| rw [integral_map | |
| (by fun_prop : Continuous (fun x => (a, characterMap q x))).measurable.aemeasurable | |
| f.continuous.aestronglyMeasurable] | |
| theorem integral_haarPhaseLaw {Z : Type*} [TopologicalSpace Z] | |
| [MeasurableSpace Z] [BorelSpace Z] (m : ℕ) (a : Z) | |
| (f : C(Z × UnitAddTorus (Fin m), ℂ)) : | |
| (∫ z, f z ∂(haarPhaseLaw m a : Measure _)) = | |
| ∫ y : UnitAddTorus (Fin m), f (a, y) := by | |
| change (∫ z, f z ∂Measure.map (fun y : UnitAddTorus (Fin m) => (a, y)) volume) = _ | |
| rw [integral_map | |
| (by fun_prop : Continuous (fun y : UnitAddTorus (Fin m) => (a, y))).measurable.aemeasurable | |
| f.continuous.aestronglyMeasurable] | |
| theorem tendsto_characterPhaseLaw | |
| {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] | |
| (q : ℕ → ℕ → ℕ) (m : 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) | |
| {a : ℕ → Z} {b : Z} (ha : Tendsto a atTop (𝓝 b)) : | |
| Tendsto (fun j => characterPhaseLaw (fun i : Fin m => q j i.val) (a j)) atTop | |
| (𝓝 (haarPhaseLaw m b)) := by | |
| apply (ProbabilityMeasure.tendsto_iff_forall_integral_rclike_tendsto ℂ).mpr | |
| intro f | |
| have h := tendsto_integral_character_phase q m hq hmono hgap ha f.toContinuousMap | |
| simpa only [← BoundedContinuousFunction.coe_toContinuousMap, | |
| integral_characterPhaseLaw, integral_haarPhaseLaw, characterMap, ContinuousMap.coe_mk] using h | |
| end CharacterLaws | |
| end Invariant233 | |
| end | |
| end OAI | |