| import OAI.LinearAlgebra.MatrixMultiplication.Duality.FamilyConstruction |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessRealization |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessRates |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.DualWitnessRealization |
|
|
| @[simp] theorem pairRate_law (pair : ReaderPair) : |
| DualFiniteRealization.pairRate hierarchy DualWitness.law pair = DualWitness.rate pair := rfl |
|
|
| end MatrixMultiplication.DualWitnessRealization |
|
|
| namespace MatrixMultiplication.DualExponentBound |
|
|
| open DualWitnessRealization |
|
|
| theorem fixedAspect_actual_families : |
| ∀ ε : ℝ, 0 < ε → Nonempty (ActualFamily fixedAspect (2 + ε)) := by |
| apply DualFiniteRealization.arbitrarily_close_families_of_exact_incident |
| hierarchy hierarchy_complete DualWitness.law approximation |
| (pow_pos (by decide : 0 < (2 : ℕ)) 288) source_supported source_unit |
| source_log_positive |
| · rw [pairRate_law, pairRate_law, source_log] |
| exact DualWitnessRates.incident_rate |
| · rw [pairRate_law, pairRate_law, pairRate_law, |
| DualWitnessRates.incident_rate, DualWitnessRates.hidden_rate] |
| have hlog : 0 < Real.log (2 : ℝ) := Real.log_pos (by norm_num) |
| have hgap := (lt_div_iff₀ hlog).mp fixedAspect_lt_dual_aspect |
| change fixedAspect * Real.log 2 < 2 * DualWitness.hiddenRate at hgap |
| nlinarith |
|
|
| theorem fixedAspect_omega_eq_two : Arithmetic.rectangularOmega ℂ fixedAspect = 2 := |
| omega_eq_two_of_arbitrarily_close_families fixedAspect_mem_unit.1 fixedAspect_actual_families |
|
|
| theorem alpha_gt : (93 : ℝ) / 200 < Arithmetic.complexAlpha := |
| alpha_gt_target_of_arbitrarily_close_families fixedAspect_actual_families |
|
|
| end MatrixMultiplication.DualExponentBound |
|
|
| end |
|
|
| end OAI |
|
|
