File size: 1,978 Bytes
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 49 50 51 52 53 54 55 56 57 | import OAI.LinearAlgebra.MatrixMultiplication.ComplexArithmetic.NaiveAlgorithm
import OAI.LinearAlgebra.MatrixMultiplication.ComplexArithmetic.LowerBound
namespace OAI
/-! Complex arithmetic programs and asymptotic matrix multiplication costs. -/
noncomputable section
namespace MatrixMultiplication.Foundation.Arithmetic
theorem AdmissibleExponent.mono {τ σ : ℝ} (hτ : AdmissibleExponent τ)
(hτσ : τ ≤ σ) : AdmissibleExponent σ := by
intro ε hε
obtain ⟨C, hC, hbound⟩ := hτ ε hε
refine ⟨C, hC, ?_⟩
intro n hn
obtain ⟨P, hP, hcost⟩ := hbound n hn
refine ⟨P, hP, hcost.trans ?_⟩
apply mul_le_mul_of_nonneg_left _ hC.le
exact Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast hn)
(add_le_add_left hτσ ε)
theorem omega_le_of_admissibleExponent {τ : ℝ} (hτ : AdmissibleExponent τ) :
omega ≤ τ :=
csInf_le admissibleExponent_bddBelow hτ
theorem omega_nonneg : 0 ≤ omega :=
le_csInf admissibleExponent_nonempty (fun _ hτ => admissibleExponent_nonneg hτ)
theorem omega_le_three : omega ≤ 3 :=
omega_le_of_admissibleExponent admissibleExponent_three
theorem omega_admissibleExponent : AdmissibleExponent omega := by
intro ε hε
have hhalf : 0 < ε / 2 := half_pos hε
obtain ⟨τ, hτ, hclose⟩ := exists_lt_of_csInf_lt admissibleExponent_nonempty
(show sInf {σ : ℝ | AdmissibleExponent σ} < omega + ε / 2 from
lt_add_of_pos_right omega hhalf)
obtain ⟨C, hC, hbound⟩ := hτ (ε / 2) hhalf
refine ⟨C, hC, ?_⟩
intro n hn
obtain ⟨P, hP, hcost⟩ := hbound n hn
refine ⟨P, hP, hcost.trans ?_⟩
apply mul_le_mul_of_nonneg_left _ hC.le
exact Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast hn) (by linarith)
theorem admissibleExponent_iff_omega_le {τ : ℝ} :
AdmissibleExponent τ ↔ omega ≤ τ :=
⟨omega_le_of_admissibleExponent, omega_admissibleExponent.mono⟩
end MatrixMultiplication.Foundation.Arithmetic
end
end OAI
|