openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /SpectralLimit.lean
| import Mathlib.Analysis.SpecialFunctions.Pow.Continuity | |
| import Mathlib.Analysis.SpecificLimits.Basic | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! | |
| # Removing positive slack from the spectral inequality | |
| The positive slack in the exponent can be sent to zero with the matrix size | |
| fixed. A bound at all positive natural sizes then removes a fixed additive | |
| error term. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open Filter Topology | |
| /-- Continuity removes the positive slack in the spectral exponent. -/ | |
| theorem spectral_bound_of_positive_slack {ν k D : ℝ} {d n : ℕ} | |
| (hν : 0 < ν) (hn : 1 ≤ n) | |
| (hbound : ∀ δ : ℝ, 0 < δ → | |
| (d : ℝ) ^ ν * (n : ℝ) ^ (ν / (ν + δ)) ≤ (n : ℝ) * k + D) : | |
| (d : ℝ) ^ ν * (n : ℝ) ≤ (n : ℝ) * k + D := by | |
| have hn0 : (n : ℝ) ≠ 0 := by | |
| have : (0 : ℝ) < n := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hn) | |
| exact this.ne' | |
| have hδ : Tendsto (fun m : ℕ => 1 / ((m : ℝ) + 1)) atTop (𝓝 0) := | |
| tendsto_one_div_add_atTop_nhds_zero_nat | |
| have hdenom : Tendsto (fun m : ℕ => ν + 1 / ((m : ℝ) + 1)) | |
| atTop (𝓝 ν) := by | |
| simpa only [add_zero] using (tendsto_const_nhds (x := ν)).add hδ | |
| have hexponent : Tendsto (fun m : ℕ => ν / (ν + 1 / ((m : ℝ) + 1))) | |
| atTop (𝓝 1) := by | |
| have h : Tendsto (fun m : ℕ => ν / (ν + 1 / ((m : ℝ) + 1))) | |
| atTop (𝓝 (ν / ν)) := by | |
| exact (tendsto_const_nhds (x := ν)).div hdenom hν.ne' | |
| rwa [div_self hν.ne'] at h | |
| have hpower : Tendsto | |
| (fun m : ℕ => (d : ℝ) ^ ν * (n : ℝ) ^ (ν / (ν + 1 / ((m : ℝ) + 1)))) | |
| atTop (𝓝 ((d : ℝ) ^ ν * (n : ℝ))) := by | |
| simpa using tendsto_const_nhds.mul | |
| ((Real.continuousAt_const_rpow hn0).tendsto.comp hexponent) | |
| exact le_of_tendsto' hpower fun m => hbound _ (by positivity) | |
| /-- A fixed additive error cannot enlarge a coefficient bounded at every | |
| positive natural size. -/ | |
| theorem coefficient_le_of_nat_mul_bound {a k D : ℝ} | |
| (hbound : ∀ n : ℕ, 1 ≤ n → a * (n : ℝ) ≤ (n : ℝ) * k + D) : | |
| a ≤ k := by | |
| by_contra h | |
| have hgap : 0 < a - k := sub_pos.mpr (lt_of_not_ge h) | |
| obtain ⟨n, hn⟩ := exists_nat_gt (max (D / (a - k)) 0) | |
| have hn0 : (0 : ℝ) < n := lt_of_le_of_lt (le_max_right _ _) hn | |
| have hn1 : 1 ≤ n := by | |
| have : 0 < n := by exact_mod_cast hn0 | |
| omega | |
| have hlarge : D < (n : ℝ) * (a - k) := | |
| (div_lt_iff₀ hgap).mp (lt_of_le_of_lt (le_max_left _ _) hn) | |
| have hsmall := hbound n hn1 | |
| nlinarith | |
| /-- Uniform spectral inequalities for all positive sizes and all positive | |
| slacks force the desired coefficient bound. -/ | |
| theorem spectral_coefficient_le_of_all_nat {ν k D : ℝ} {d : ℕ} | |
| (hν : 0 < ν) | |
| (hbound : ∀ n : ℕ, 1 ≤ n → ∀ δ : ℝ, 0 < δ → | |
| (d : ℝ) ^ ν * (n : ℝ) ^ (ν / (ν + δ)) ≤ (n : ℝ) * k + D) : | |
| (d : ℝ) ^ ν ≤ k := by | |
| apply coefficient_le_of_nat_mul_bound | |
| intro n hn | |
| exact spectral_bound_of_positive_slack hν hn (hbound n hn) | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |