openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexArithmetic /Exponent.lean
| 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 | |