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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.AuxiliarySeparation.Character.Existence
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.CharacterRounding
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Polynomial.Inequalities
namespace OAI
/-!
# The exact-rank exponent bound
Detecting characters exist by Appendix A. The determinant and three-sector
polynomial constructions bound the sum of their dot-product exponents by
`9/4`. Integer rounding then gives the same bound for the exact-rank exponent.
-/
namespace MatrixMultiplication.AuxiliarySeparation
/-- The polynomial inequalities bound the actual exact-rank exponent. -/
theorem exactRankExponent_le_nine_quarters : exactRankExponent ≤ (9 : ℝ) / 4 := by
apply exactRankExponent_le_nine_quarters_of_characters
· intro d hd k hk
exact exists_detecting_character hd hk
· intro χ d hd
rw [χ.value_matrixMultiplication (by omega)]
exact Real.rpow_le_rpow_of_exponent_le
(by exact_mod_cast (show 1 ≤ d by omega)) χ.exponent_sum_le_nine_quarters
end MatrixMultiplication.AuxiliarySeparation
end OAI
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