openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Arithmetic /RankBound.lean
| 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 | |