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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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | 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
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