| import OAI.Analysis.IrrationalRotation.Couplings.CircleDigitCoupling |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace Invariant233.CircleDigitCoupling |
| open CouplingMarginalBounds CouplingMixture OneCoordinateCoupling |
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|
| variable {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) |
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| include hm ha ha' hb hb' in |
| lemma endpointLaw_eq_density (side : Bool) (l : Label) : |
| endpointLaw side m a b l = baseLaw.withDensity |
| (fun x => ENNReal.ofReal (endpointDensity side m a b l x)) := by |
| have hm0 : 0 < m := by linarith |
| cases l with |
| | none => simp [endpointLaw, endpointDensity] |
| | some i => |
| cases side |
| · exact leftLaw_eq_withDensity (by linarith : 2 ≤ 2 * m) |
| (width_pos hm0 ha) ha' (width_pos hm0 hb) hb' i |
| · exact rightLaw_eq_withDensity (by linarith : 2 ≤ 2 * m) |
| (width_pos hm0 ha) ha' (width_pos hm0 hb) hb' i |
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| lemma endpointDensity_measurable (side : Bool) (m a b : ℝ) (l : Label) : |
| Measurable (endpointDensity side m a b l) := by |
| cases l with |
| | none => exact measurable_const |
| | some i => cases side <;> fin_cases i <;> exact intervalDensity_measurable _ _ |
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| lemma endpointDensity_nonneg (side : Bool) (m a b : ℝ) (l : Label) (x : ℝ) : |
| 0 ≤ endpointDensity side m a b l x := by |
| cases l with |
| | none => norm_num [endpointDensity] |
| | some i => cases side <;> fin_cases i <;> exact intervalDensity_nonneg _ _ _ |
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| include hκ0 hκ hm ha ha' hb hb' ht0 ht in |
| lemma marginalDensity_bounds (side : Bool) (x : ℝ) : |
| 49 / 50 ≤ marginalDensity side κ m a b t x ∧ |
| marginalDensity side κ m a b t x ≤ 3 + 8 * κ ∧ |
| marginalDensity side κ m a b t x < 10 := by |
| have hm0 : 0 < m := by linarith |
| have ha0 := width_pos hm0 ha |
| have hb0 := width_pos hm0 hb |
| have hd (i : Fin 5) : 0 ≤ endpointDensity side m a b (some i) x ∧ |
| endpointDensity side m a b (some i) x ≤ componentVarianceBounds (2 * m) a b i := by |
| cases side |
| · exact leftDensity_bound (by linarith) ha0 ha' hb0 hb' i x |
| · exact rightDensity_bound (by linarith) ha0 ha' hb0 hb' i x |
| have hρa := rho_div_width_le hκ0 hm0 ha |
| have hρb := rho_div_width_le hκ0 hm0 hb |
| have hρa0 := rho_nonneg (m := m) (w := a) hκ0 |
| have hρb0 := rho_nonneg (m := m) (w := b) hκ0 |
| have hq := diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht |
| have htag0 : 0 ≤ t / m := div_nonneg ht0 hm0.le |
| have htag : (t / m) * endpointDensity side m a b (some 4) x ≤ 2 := by |
| calc |
| _ ≤ (t / m) * (2 * m) := mul_le_mul_of_nonneg_left (hd 4).2 htag0 |
| _ = 2 * t := by field_simp |
| _ ≤ 2 := by linarith |
| have hw (w : ℝ) (v : ℝ) (hw0 : 0 < w) (hρ : rho κ m w / w ≤ κ) |
| (hv : v ≤ 2 / w) : rho κ m w * v ≤ 2 * κ := by |
| calc |
| _ ≤ rho κ m w * (2 / w) := mul_le_mul_of_nonneg_left hv (rho_nonneg hκ0) |
| _ = 2 * (rho κ m w / w) := by ring |
| _ ≤ _ := by linarith |
| have h0 := hw a _ ha0 hρa (hd 0).2 |
| have h1 := hw a _ ha0 hρa (hd 1).2 |
| have h2 := hw b _ hb0 hρb (hd 2).2 |
| have h3 := hw b _ hb0 hρb (hd 3).2 |
| have hp0 (i : Fin 5) : 0 ≤ componentProbabilities κ m a b t i * |
| endpointDensity side m a b (some i) x := |
| mul_nonneg (componentProbabilities_nonneg hκ0 hm0 ht0 i) (hd i).1 |
| have hlo : 49 / 50 ≤ marginalDensity side κ m a b t x := by |
| exact hq.1.trans (le_add_of_nonneg_right (Finset.sum_nonneg (fun i _ => hp0 i))) |
| have hup : marginalDensity side κ m a b t x ≤ 3 + 8 * κ := by |
| unfold marginalDensity |
| simp only [componentProbabilities, Fin.sum_univ_succ, Matrix.cons_val_zero, |
| Matrix.cons_val_succ, Fin.sum_univ_zero, add_zero] |
| change diagonalMass κ m a b t + (rho κ m a * endpointDensity side m a b (some 0) x + |
| (rho κ m a * endpointDensity side m a b (some 1) x + |
| (rho κ m b * endpointDensity side m a b (some 2) x + |
| (rho κ m b * endpointDensity side m a b (some 3) x + |
| t / m * endpointDensity side m a b (some 4) x)))) ≤ _ |
| linarith [hq.2] |
| exact ⟨hlo, hup, by linarith⟩ |
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|
| include hκ0 hκ hm ha ha' hb hb' ht0 ht in |
| lemma ofReal_marginalDensity (side : Bool) (x : ℝ) : |
| ENNReal.ofReal (marginalDensity side κ m a b t x) = |
| labelMass κ m a b t none + ∑ i : Fin 5, |
| labelMass κ m a b t (some i) * |
| ENNReal.ofReal (endpointDensity side m a b (some i) x) := by |
| have hm0 : 0 < m := by linarith |
| apply ofReal_mixture_density |
| · exact le_trans (by norm_num) (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1 |
| · exact componentProbabilities_nonneg hκ0 hm0 ht0 |
| · intro i y |
| exact endpointDensity_nonneg side m a b (some i) y |
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| include hκ0 hκ hm ha ha' hb hb' ht0 ht in |
| lemma digitWeight_total_eq_integral (side : Bool) (r : ℕ) (d : Fin r) : |
| (∑ l, digitWeight side r κ m a b t l d) = |
| ∫⁻ x in bin r d, ENNReal.ofReal (marginalDensity side κ m a b t x) ∂baseLaw := by |
| have h := withDensity_finite_mixture baseLaw (labelMass κ m a b t none) |
| (fun i : Fin 5 => labelMass κ m a b t (some i)) |
| (fun i x => ENNReal.ofReal (endpointDensity side m a b (some i) x)) |
| (fun i => (endpointDensity_measurable side m a b (some i)).ennreal_ofReal) |
| simp_rw [← ofReal_marginalDensity hκ0 hκ hm ha ha' hb hb' ht0 ht side] at h |
| have heval := congrArg (fun ν : Measure ℝ => ν (bin r d)) h |
| rw [withDensity_apply _ (show MeasurableSet (bin r d) from measurableSet_Ico)] at heval |
| rw [heval, Fintype.sum_option] |
| simp only [Measure.add_apply, Measure.smul_apply, Measure.finsetSum_apply, smul_eq_mul] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [← endpointLaw_eq_density hm ha ha' hb hb' side (some i)] |
| rfl |
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| include hκ0 hκ hm ha ha' hb hb' ht0 ht in |
| lemma digitWeight_total_le_ten (side : Bool) (r : ℕ) (d : Fin r) : |
| (∑ l, digitWeight side r κ m a b t l d) ≤ 10 * baseLaw (bin r d) := by |
| rw [digitWeight_total_eq_integral hκ0 hκ hm ha ha' hb hb' ht0 ht] |
| calc |
| _ ≤ ∫⁻ _x in bin r d, (10 : ℝ≥0∞) ∂baseLaw := by |
| apply lintegral_mono |
| intro x |
| exact le_trans (ENNReal.ofReal_le_ofReal |
| (marginalDensity_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht side x).2.2.le) |
| (by norm_num) |
| _ = _ := by simp |
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| end Invariant233.CircleDigitCoupling |
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| end |
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| end OAI |
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