| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Main | |
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedRectangular | |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.ExponentBound | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Conditional | |
| import OAI.LinearAlgebra.MatrixMultiplication.Numerical.AllFieldCertificatesReplayIntegration | |
| namespace OAI | |
| /-! Arithmetic complexity bounds for square and rectangular matrix multiplication. -/ | |
| namespace MatrixMultiplication | |
| /-- The complex square matrix multiplication exponent is at most `9/4`. -/ | |
| theorem complex_omega_le_nine_quarters : | |
| Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := | |
| AuxiliarySeparation.omega_le_nine_quarters | |
| /-- The complex dual matrix multiplication exponent is greater than `0.465`. -/ | |
| theorem complex_alpha_gt_93_div_200 : | |
| (93 : ℝ) / 200 < Arithmetic.complexAlpha := | |
| DualExponentBound.alpha_gt | |
| /-- Multiplication of `n × n^0.709` by `n^0.709 × n` complex matrices has | |
| arithmetic exponent less than `2.092`. -/ | |
| theorem complex_rectangular_omega_lt_523_div_250 : | |
| Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := | |
| CW75ReorderedRectangular.rectangularOmega_lt_target | |
| end MatrixMultiplication | |
| end OAI | |