openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /CapacityAlgebra.lean
| 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 | |