openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Main.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.Exponent | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.RankBound | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Convolution.RankLowerBound | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Determinant.Bounds | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.Floor | |
| namespace OAI | |
| /-! | |
| # Matrix multiplication via auxiliary separation and polynomial multiplication | |
| Theorem 1.1 gives a uniform arithmetic-operation bound for complex square | |
| matrix multiplication, and consequently an upper bound of `9/4` on its exponent. | |
| The two conclusions are stated separately in the existing arithmetic model. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- Theorem 1.1: for every positive exponent slack, one constant bounds the cost | |
| of correct complex matrix multiplication programs at every positive size. -/ | |
| theorem matrix_multiplication_cost_le (ε : ℝ) (hε : 0 < ε) : | |
| ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ, 1 ≤ n → | |
| ∃ P : Arithmetic.MatrixAlgorithm ℂ n n n, | |
| P.Correct ∧ (P.cost : ℝ) ≤ C * (n : ℝ) ^ ((9 : ℝ) / 4 + ε) := | |
| arithmeticComplexity_of_exactRankExponent_le exactRankExponent_le_nine_quarters ε hε | |
| /-- The complex arithmetic matrix multiplication exponent is at most `9/4`. -/ | |
| theorem omega_le_nine_quarters : Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := | |
| omega_le_exactRankExponent.trans exactRankExponent_le_nine_quarters | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |