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cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.EntropyMax
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
namespace MatrixMultiplication.JointCapacityAlgebra
open scoped BigOperators
theorem total_sub_max_degrees (A X Y Z : ℝ) :
A - max (A - X) (max (A - Y) (A - Z)) = min X (min Y Z) := by
simp only [max_def, min_def]
split_ifs <;> linarith
theorem weighted_joint_capacity {H : Type*} [Fintype H]
(mass entropy : H → ℝ) (capacity : H → Fin 3 → ℝ) :
(∑ h, mass h * entropy h) -
max (∑ h, mass h * (entropy h - capacity h 0))
(max (∑ h, mass h * (entropy h - capacity h 1))
(∑ h, mass h * (entropy h - capacity h 2))) =
min (∑ h, mass h * capacity h 0)
(min (∑ h, mass h * capacity h 1) (∑ h, mass h * capacity h 2)) := by
simp_rw [mul_sub, Finset.sum_sub_distrib]
exact total_sub_max_degrees _ _ _ _
end MatrixMultiplication.JointCapacityAlgebra
end OAI
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