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Theorem 4.2: checkerboard lower bound for Chatterjee's xi

theorem Papers.Rockel2025Approximation.mtp2_isCI {C : ProbabilityTheory.Copula 2} (hC : C.HasMTP2Density) : C.IsCI

The source's MTP2 density implies conditional increase in both directions.

theorem Papers.Rockel2025Approximation.checkerboard_xi_le_of_isCI {n : ℕ} (C : ProbabilityTheory.Copula 2) (hC : C.IsCI) (m : ℕ) (hm : 0 < m) (Q : ProbabilityTheory.Copula.IntervalPartition n) : (C.cellMass (ProbabilityTheory.Copula.IntervalPartition.uniform m hm) Q).checkerboard.chatterjeeXi ≤ C.chatterjeeXi

The comparison holds for CI copulas with any response partition.

theorem Papers.Rockel2025Approximation.checkerboard_xi_le_of_mtp2 (C : ProbabilityTheory.Copula 2) (hC : C.HasMTP2Density) (m n : ℕ) (hm : 0 < m) (hn : 0 < n) : (C.cellMass (ProbabilityTheory.Copula.IntervalPartition.uniform m hm) (ProbabilityTheory.Copula.IntervalPartition.uniform n hn)).checkerboard.chatterjeeXi ≤ C.chatterjeeXi

Theorem 4.2, for every positive rectangular grid and its actual source cell masses.