File size: 2,354 Bytes
2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Algebra.DepthFive.FiniteOperator
import OAI.Algebra.DepthFive.LowerSchedule
import OAI.Algebra.DepthFive.AnalyticParameters
import OAI.Algebra.DepthFive.BinomialRatios
namespace OAI
/-! Common, concrete data for the two sides of the IMM rank comparison. -/
noncomputable section
namespace Problem335.LowerParameters
/-- The source dimension for the configured bidegree rank measure. -/
def rankDimension (n : ℕ) : ℝ :=
(homogeneousDim (v n) (a n) : ℝ) * (homogeneousDim (u n) (b n) : ℝ)
/-- The common path-count and occupation-mean factor in the two trace moments. -/
def rankMomentScale (n : ℕ) : ℝ :=
(n : ℝ) ^ (n - 1) * ((a n : ℝ) / (v n : ℝ)) ^ k n *
(1 + (b n : ℝ) / (u n : ℝ)) ^ m n
/-- The actual rank measure, extended by zero outside the range where the
balanced schedule is used. The small values do not enter the eventual bounds. -/
def rankData (n : ℕ) (p : MvPolynomial (Fin n × Fin n × Fin n) ℂ) : ℝ :=
if hn : 4 ≤ n then
(bidegreeRank (fun x : Fin n × Fin n × Fin n => balancedLayer hn x.1)
(a n) (b n) (k n) (m n) p : ℝ)
else 0
theorem rankData_eq {n : ℕ} (hn : 4 ≤ n)
(p : MvPolynomial (Fin n × Fin n × Fin n) ℂ) :
rankData n p =
(bidegreeRank (fun x : Fin n × Fin n × Fin n => balancedLayer hn x.1)
(a n) (b n) (k n) (m n) p : ℝ) := by
simp [rankData, hn]
theorem rankData_nonneg (n : ℕ)
(p : MvPolynomial (Fin n × Fin n × Fin n) ℂ) : 0 ≤ rankData n p := by
unfold rankData
split_ifs <;> positivity
theorem rankDimension_nonneg (n : ℕ) : 0 ≤ rankDimension n := by
unfold rankDimension
positivity
theorem rankDimension_pos {n : ℕ} (hn : 4 ≤ n) : 0 < rankDimension n := by
have hv : 0 < homogeneousDim (v n) (a n) :=
homogeneousDim_pos (v_pos hn) (a n)
have hu : 0 < homogeneousDim (u n) (b n) :=
homogeneousDim_pos (u_pos hn) (b n)
unfold rankDimension
positivity
theorem rankMomentScale_pos {n : ℕ} (hn : 4 ≤ n) : 0 < rankMomentScale n := by
have hn' : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
have ha : (0 : ℝ) < a n := by exact_mod_cast a_pos hn
have hv : (0 : ℝ) < v n := by exact_mod_cast v_pos hn
have hu : (0 : ℝ) < u n := by exact_mod_cast u_pos hn
unfold rankMomentScale
positivity
end Problem335.LowerParameters
end
end OAI
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