openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Arithmetic /ExponentComparison.lean
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! | |
| # Comparing power growth | |
| The last scalar step of the auxiliary-separation argument compares a | |
| fourth-power lower bound for the cube of the diagonal profile with its | |
| dimension upper bound. Powers of two suffice to compare the exponents, so | |
| the argument needs no asymptotic estimates or approximation of constants. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- A uniform bound on all positive natural inputs forces the exponent on | |
| the left to be no larger than the exponent on the right. -/ | |
| theorem rpow_exponent_le_of_nat_bound {a b C : ℝ} | |
| (hbound : ∀ n : ℕ, 1 ≤ n → (n : ℝ) ^ a ≤ C * (n : ℝ) ^ b) : | |
| a ≤ b := by | |
| have hC : 0 < C := by | |
| have h := hbound 1 (by omega) | |
| norm_num at h | |
| linarith | |
| by_contra hab | |
| have hab' : 0 < a - b := sub_pos.mpr (lt_of_not_ge hab) | |
| have hlog2 : 0 < Real.log (2 : ℝ) := Real.log_pos (by norm_num) | |
| have hp : 0 < (a - b) * Real.log 2 := mul_pos hab' hlog2 | |
| obtain ⟨k, hk⟩ := exists_nat_gt (Real.log C / ((a - b) * Real.log 2)) | |
| have hk' : Real.log C < (k : ℝ) * ((a - b) * Real.log 2) := | |
| (div_lt_iff₀ hp).mp hk | |
| have hn : (0 : ℝ) < (2 ^ k : ℕ) := by positivity | |
| have h := Real.log_le_log (Real.rpow_pos_of_pos hn a) | |
| (hbound (2 ^ k) (Nat.one_le_iff_ne_zero.mpr (by positivity))) | |
| rw [Real.log_mul hC.ne' (Real.rpow_pos_of_pos hn b).ne', | |
| Real.log_rpow hn a, Real.log_rpow hn b] at h | |
| simp only [Nat.cast_pow, Nat.cast_ofNat, Real.log_pow] at h | |
| nlinarith | |
| /-- The diagonal lower bound and the dimension upper bound force the scalar | |
| exponent to be at most `3/4`. -/ | |
| theorem diagonal_exponent_le_three_quarters {D : ℕ → ℝ} {t : ℝ} | |
| (ht : 0 < t) | |
| (hnonneg : ∀ n : ℕ, 1 ≤ n → 0 ≤ D n) | |
| (hlower : ∀ n : ℕ, 1 ≤ n → (n : ℝ) ^ 4 ≤ D n ^ 3) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → D n ≤ (2 * (n : ℝ) - 1) ^ (1 / t)) : | |
| t ≤ 3 / 4 := by | |
| have hexp : 0 ≤ 1 / t := le_of_lt (one_div_pos.mpr ht) | |
| have hpower : ∀ n : ℕ, 1 ≤ n → | |
| (n : ℝ) ^ (4 : ℝ) ≤ | |
| (2 : ℝ) ^ ((1 / t) * 3) * (n : ℝ) ^ ((1 / t) * 3) := by | |
| intro n hn | |
| have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn | |
| have hbase : 0 ≤ 2 * (n : ℝ) - 1 := by linarith | |
| have hu : D n ≤ (2 * (n : ℝ)) ^ (1 / t) := | |
| (hupper n hn).trans (Real.rpow_le_rpow hbase (by linarith) hexp) | |
| calc | |
| (n : ℝ) ^ (4 : ℝ) = (n : ℝ) ^ 4 := by norm_num | |
| _ ≤ D n ^ 3 := hlower n hn | |
| _ ≤ ((2 * (n : ℝ)) ^ (1 / t)) ^ 3 := | |
| pow_le_pow_left₀ (hnonneg n hn) hu 3 | |
| _ = (2 * (n : ℝ)) ^ ((1 / t) * 3) := by | |
| simpa only [Nat.cast_ofNat] using | |
| (Real.rpow_mul_natCast (show 0 ≤ 2 * (n : ℝ) by positivity) (1 / t) 3).symm | |
| _ = (2 : ℝ) ^ ((1 / t) * 3) * (n : ℝ) ^ ((1 / t) * 3) := | |
| Real.mul_rpow (by norm_num) (by positivity) | |
| have hcompare : (4 : ℝ) ≤ (1 / t) * 3 := | |
| rpow_exponent_le_of_nat_bound hpower | |
| have hmul := mul_le_mul_of_nonneg_right hcompare (le_of_lt ht) | |
| have hcancel : ((1 / t) * 3) * t = (3 : ℝ) := by | |
| field_simp | |
| rw [hcancel] at hmul | |
| linarith | |
| /-- The scalar comparison in the normalization used for the matrix | |
| multiplication exponent. -/ | |
| theorem three_mul_diagonal_exponent_le_nine_quarters {D : ℕ → ℝ} {t : ℝ} | |
| (ht : 0 < t) | |
| (hnonneg : ∀ n : ℕ, 1 ≤ n → 0 ≤ D n) | |
| (hlower : ∀ n : ℕ, 1 ≤ n → (n : ℝ) ^ 4 ≤ D n ^ 3) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → D n ≤ (2 * (n : ℝ) - 1) ^ (1 / t)) : | |
| 3 * t ≤ 9 / 4 := by | |
| have h := diagonal_exponent_le_three_quarters ht hnonneg hlower hupper | |
| linarith | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |