| import OAI.Analysis.IrrationalRotation.Sampling.CircleDigitDensity |
| import OAI.Analysis.IrrationalRotation.Sampling.CircleDigitDisintegration |
| import OAI.Analysis.IrrationalRotation.Analysis.QuantizedDensity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace Invariant233.CircleDigitCoupling |
| open CouplingMarginalBounds CouplingMixture OneCoordinateCoupling |
|
|
| lemma endpointDensity_memLp (side : Bool) (m a b : ℝ) (l : Label) : |
| MemLp (endpointDensity side m a b l) 2 baseLaw := by |
| cases l with |
| | none => exact memLp_const 1 |
| | some i => cases side <;> fin_cases i <;> exact intervalDensity_memLp _ _ 2 |
|
|
| lemma marginalDensity_memLp (side : Bool) (κ m a b t : ℝ) : |
| MemLp (marginalDensity side κ m a b t) 2 baseLaw := by |
| apply (memLp_const (diagonalMass κ m a b t)).add |
| exact memLp_finsetSum Finset.univ (fun i _ => |
| (endpointDensity_memLp side m a b (some i)).const_mul _) |
|
|
| lemma endpointDensity_integral {m a b : ℝ} (hm : 100 ≤ m) |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) (side : Bool) (l : Label) : |
| (∫ x, endpointDensity side m a b l x ∂baseLaw) = 1 := by |
| have := endpointLaw_isProbabilityMeasure hm ha ha' hb hb' side l |
| have h := congrArg (fun ν : Measure ℝ => ν univ) |
| (endpointLaw_eq_density hm ha ha' hb hb' side l) |
| rw [measure_univ, withDensity_apply _ MeasurableSet.univ, |
| Measure.restrict_univ, ← ofReal_integral_eq_lintegral_ofReal |
| ((endpointDensity_memLp side m a b l).integrable (by norm_num)) |
| (Filter.Eventually.of_forall (endpointDensity_nonneg side m a b l))] at h |
| have hn : 0 ≤ ∫ x, endpointDensity side m a b l x ∂baseLaw := |
| integral_nonneg (endpointDensity_nonneg side m a b l) |
| exact (ENNReal.ofReal_eq_ofReal_iff hn (by norm_num)).mp (by simpa using h.symm) |
|
|
| lemma marginalDensity_integral {κ m a b t : ℝ} (hm : 100 ≤ m) |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) (side : Bool) : |
| (∫ x, marginalDensity side κ m a b t x ∂baseLaw) = 1 := by |
| have hi (i : Fin 5) : Integrable |
| (fun x => componentProbabilities κ m a b t i * endpointDensity side m a b (some i) x) |
| baseLaw := |
| ((endpointDensity_memLp side m a b (some i)).integrable (by norm_num)).const_mul _ |
| unfold marginalDensity |
| rw [integral_add (integrable_const _) (integrable_finsetSum Finset.univ (fun i _ => hi i)), |
| integral_finsetSum Finset.univ (fun i _ => hi i)] |
| simp only [integral_const, probReal_univ, one_smul, integral_const_mul, |
| endpointDensity_integral hm ha ha' hb hb' side, mul_one] |
| simp only [componentProbabilities, Fin.sum_univ_succ, Matrix.cons_val_zero, |
| Matrix.cons_val_succ, Fin.sum_univ_zero, add_zero, diagonalMass] |
| ring |
|
|
| lemma marginalDensity_energy {κ m a b t : ℝ} (hm : 100 ≤ m) |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) |
| (ht0 : 0 ≤ t) (ht : t ≤ 1) (side : Bool) : |
| (∫ x, (marginalDensity side κ m a b t x - 1) ^ 2 ∂baseLaw) ≤ |
| 50 * (κ ^ 2 / m + t / m) := by |
| have hm0 : 0 < m := by linarith |
| have ha0 := width_pos hm0 ha |
| have hb0 := width_pos hm0 hb |
| have hf (i : Fin 5) : MemLp |
| (fun x => endpointDensity side m a b (some i) x - 1) 2 baseLaw := |
| (endpointDensity_memLp side m a b (some i)).sub (memLp_const 1) |
| have hv (i : Fin 5) : |
| (∫ x, (endpointDensity side m a b (some i) x - 1) ^ 2 ∂baseLaw) ≤ |
| componentVarianceBounds (2 * m) a b i := by |
| cases side |
| · exact leftDensity_variance_bound (by linarith) ha0 ha' hb0 hb' i |
| · exact rightDensity_variance_bound (by linarith) ha0 ha' hb0 hb' i |
| have hcs := integral_sq_weighted_sum_le hf (componentProbabilities κ m a b t) |
| have hs : (∑ i, componentProbabilities κ m a b t i ^ 2 * |
| ∫ x, (endpointDensity side m a b (some i) x - 1) ^ 2 ∂baseLaw) ≤ |
| ∑ i, componentProbabilities κ m a b t i ^ 2 * |
| componentVarianceBounds (2 * m) a b i := |
| Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hv i) (sq_nonneg _)) |
| have he : (∑ i, componentProbabilities κ m a b t i ^ 2 * |
| componentVarianceBounds (2 * m) a b i) = 8 * κ ^ 2 / m + 2 * t ^ 2 / m := by |
| simp only [componentProbabilities, componentVarianceBounds, Fin.sum_univ_succ, |
| Matrix.cons_val_zero, Matrix.cons_val_succ, Fin.sum_univ_zero, add_zero] |
| have hra := rho_sq_div_width (κ := κ) hm0 ha0 |
| have hrb := rho_sq_div_width (κ := κ) hm0 hb0 |
| have htag : (t / m) ^ 2 * (2 * m) = 2 * t ^ 2 / m := by field_simp |
| rw [htag] |
| simp only [div_eq_mul_inv] at hra hrb ⊢ |
| nlinarith |
| have ht2 : t ^ 2 / m ≤ t / m := |
| div_le_div_of_nonneg_right (by nlinarith) hm0.le |
| have hκ2 : 0 ≤ κ ^ 2 / m := div_nonneg (sq_nonneg _) hm0.le |
| have htdiv : 0 ≤ t / m := div_nonneg ht0 hm0.le |
| simp only [Fintype.card_fin] at hcs |
| calc |
| _ = ∫ x, (∑ i, componentProbabilities κ m a b t i * |
| (endpointDensity side m a b (some i) x - 1)) ^ 2 ∂baseLaw := by |
| congr 1 |
| funext x |
| rw [marginalDensity, OneCoordinateCoupling.marginalDensity_sub_one] |
| _ ≤ 5 * (∑ i, componentProbabilities κ m a b t i ^ 2 * |
| componentVarianceBounds (2 * m) a b i) := |
| hcs.trans (mul_le_mul_of_nonneg_left hs (by norm_num)) |
| _ ≤ _ := by |
| rw [he] |
| simp only [div_eq_mul_inv] at ht2 hκ2 htdiv ⊢ |
| nlinarith |
|
|
| lemma rawLaw_map_endpoint {κ m a b t : ℝ} |
| (hκ0 : 0 ≤ κ) (hκ : κ ≤ 1 / 100) (hm : 100 ≤ m) |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) |
| (ht0 : 0 ≤ t) (ht : t ≤ 1) (side : Bool) : |
| (rawLaw κ m a b t).map (endpoint side) = |
| baseLaw.withDensity (fun x => ENNReal.ofReal (marginalDensity side κ m a b t x)) := by |
| have hm0 : 0 < m := by linarith |
| have ha0 := width_pos hm0 ha |
| have hb0 := width_pos hm0 hb |
| have hq0 : 0 ≤ diagonalMass κ m a b t := |
| (by norm_num : (0 : ℝ) ≤ 49 / 50).trans |
| (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1 |
| have : ∀ i, IsProbabilityMeasure (leftLaw (2 * m) a b i) := |
| leftLaw_isProbabilityMeasure (by linarith) ha0 ha' hb0 hb' |
| have : ∀ i, IsProbabilityMeasure (rightLaw (2 * m) a b i) := |
| rightLaw_isProbabilityMeasure (by linarith) ha0 ha' hb0 hb' |
| cases side |
| · change (rawLaw κ m a b t).map (Prod.fst ∘ Prod.snd) = _ |
| rw [← Measure.map_map measurable_fst measurable_snd] |
| unfold rawLaw |
| rw [labelledMixture_map_snd] |
| exact mixture_fst_real_density baseLaw _ _ _ _ |
| (leftDensity (2 * m) a b) hq0 (componentProbabilities_nonneg hκ0 hm0 ht0) |
| (fun i x => (leftDensity_bound (by linarith) ha0 ha' hb0 hb' i x).1) |
| (fun i => by fin_cases i <;> exact intervalDensity_measurable _ _) |
| (leftLaw_eq_withDensity (by linarith) ha0 ha' hb0 hb') |
| · change (rawLaw κ m a b t).map (Prod.snd ∘ Prod.snd) = _ |
| rw [← Measure.map_map measurable_snd measurable_snd] |
| unfold rawLaw |
| rw [labelledMixture_map_snd] |
| exact mixture_snd_real_density baseLaw _ _ _ _ |
| (rightDensity (2 * m) a b) hq0 (componentProbabilities_nonneg hκ0 hm0 ht0) |
| (fun i x => (rightDensity_bound (by linarith) ha0 ha' hb0 hb' i x).1) |
| (fun i => by fin_cases i <;> exact intervalDensity_measurable _ _) |
| (rightLaw_eq_withDensity (by linarith) ha0 ha' hb0 hb') |
|
|
| end Invariant233.CircleDigitCoupling |
|
|
| end |
|
|
| end OAI |
|
|
