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Exact Chatterjee xi of the Cuadras–Augé family

theorem Verification.integral_cuadrasConditional_sq_response (δ u : ↑unitInterval) (hu : 0 < u) : ∫ (v : ↑unitInterval), cuadrasConditional δ u v ^ 2 = (1 - ↑δ) ^ 2 * ↑u ^ (3 - 2 * ↑δ) / 3 + (1 - ↑u ^ (3 - 2 * ↑δ)) / (3 - 2 * ↑δ)

theorem Verification.cuadrasAuge_chatterjeeXi (δ : ↑unitInterval) : (ProbabilityTheory.Copula.cuadrasAuge δ).chatterjeeXi = ↑δ ^ 2 / (2 - ↑δ)