| import OAI.NumberTheory.DukePrimeDegree.Entropy.RootInformationBound | |
| import OAI.NumberTheory.DukePrimeDegree.Dynamics.RootInvariantLeaves | |
| namespace OAI | |
| /-! # Mean root coefficients and their elementary properties | |
| The coefficient is the mean leaf information at unit scale. Diagonal | |
| covariance makes real-time mean information linear in time. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology ENNReal | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| /-- The actual mean information at unit scale, with zero diagonal. -/ | |
| def canonicalRootCoefficient (d : ℕ) (μ : Measure (X (d + 1))) [IsFiniteMeasure μ] | |
| (i j : Fin (d + 1)) : ℝ := | |
| if hij : i = j then 0 else (rootInformationMean hij μ 1).toReal | |
| variable (d : ℕ) (μ : Measure (X (d + 1))) [IsProbabilityMeasure μ] | |
| (hA : ∀ u : LogCoordinates d, μ.map (fun x => rightAction x (diagonalFlow u)) = μ) | |
| lemma canonicalRootCoefficient_diagonal (i : Fin (d + 1)) : | |
| canonicalRootCoefficient d μ i i = 0 := by simp [canonicalRootCoefficient] | |
| lemma canonicalRootCoefficient_nonneg (i j : Fin (d + 1)) : | |
| 0 ≤ canonicalRootCoefficient d μ i j := by | |
| unfold canonicalRootCoefficient | |
| split_ifs | |
| · exact le_rfl | |
| · exact ENNReal.toReal_nonneg | |
| include hA | |
| lemma canonicalRootCoefficient_le_one (i j : Fin (d + 1)) : | |
| canonicalRootCoefficient d μ i j ≤ 1 := by | |
| unfold canonicalRootCoefficient | |
| split_ifs with hij | |
| · exact zero_le_one | |
| · exact (ENNReal.toReal_mono ENNReal.one_ne_top | |
| (rootInformationMean_le_one d hij μ hA)).trans_eq ENNReal.toReal_one | |
| lemma canonicalRootCoefficient_zero_iff (i j : Fin (d + 1)) (hij : i ≠ j) : | |
| canonicalRootCoefficient d μ i j = 0 ↔ | |
| ∀ᵐ x ∂μ, intrinsicRootLeaf hij μ x = Measure.dirac 0 := by | |
| rw [canonicalRootCoefficient, dite_eq_right hij, ENNReal.toReal_eq_zero_iff] | |
| have hfin : rootInformationMean hij μ 1 ≠ ∞ := | |
| ne_of_lt ((rootInformationMean_le_one d hij μ hA).trans_lt ENNReal.one_lt_top) | |
| simp only [hfin, or_false] | |
| exact rootInformationMean_eq_zero_iff_dirac d hij μ hA | |
| lemma canonicalRootCoefficient_one_of_invariant (i j : Fin (d + 1)) (hij : i ≠ j) | |
| (hinv : ∀ t : ℝ, μ.map (fun x => rightAction x | |
| (Matrix.SpecialLinearGroup.transvection hij t)) = μ) : | |
| canonicalRootCoefficient d μ i j = 1 := by | |
| rw [canonicalRootCoefficient, dite_eq_right hij, rootInformationMean_of_invariant d hij μ hA hinv] | |
| norm_num | |
| /-- Actual real-time mean information is the root coefficient times time. -/ | |
| lemma rootInformationMean_eq_coefficient (i j : Fin (d + 1)) (hij : i ≠ j) | |
| {t : ℝ} (ht : 0 ≤ t) : | |
| rootInformationMean hij μ t = | |
| ENNReal.ofReal (t * canonicalRootCoefficient d μ i j) := by | |
| have hfin : rootInformationMean hij μ 1 ≠ ∞ := | |
| ne_of_lt ((rootInformationMean_le_one d hij μ hA).trans_lt ENNReal.one_lt_top) | |
| rw [rootInformationMean_real d hij μ hA ht, canonicalRootCoefficient, dite_eq_right hij, | |
| ENNReal.ofReal_mul ht, ENNReal.ofReal_toReal hfin] | |
| end DukePrimeDegree | |
| end | |
| end OAI | |