openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /PolynomialOverhead.lean
| import Mathlib.Analysis.SpecificLimits.Normed | |
| import Mathlib.Analysis.SpecialFunctions.Log.Basic | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! | |
| # Removing polynomial overhead from exponential inequalities | |
| The interpolation argument in Section 4 bounds tensor powers with an extra | |
| linear factor. Such a factor does not change their exponential growth rate. | |
| The statements below include eventual inequalities and constant factors, so | |
| the same argument also removes fixed counting losses in entropy limits. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open Filter | |
| open scoped Topology | |
| /-- An eventual linear prefactor cannot increase an exponential growth rate. -/ | |
| theorem le_of_eventually_pow_le_linear_mul_pow {a b K C : ℝ} | |
| (ha : 0 ≤ a) | |
| (h : ∀ᶠ j : ℕ in atTop, b ^ j ≤ ((j : ℝ) * K + C) * a ^ j) : | |
| b ≤ a := by | |
| by_contra hba | |
| have hab : a < b := lt_of_not_ge hba | |
| have hb : 0 < b := lt_of_le_of_lt ha hab | |
| have hr₀ : 0 ≤ a / b := div_nonneg ha hb.le | |
| have hr₁ : a / b < 1 := (div_lt_one hb).2 hab | |
| have hlinear := (tendsto_self_mul_const_pow_of_lt_one hr₀ hr₁).mul_const K | |
| have hconstant := (tendsto_pow_atTop_nhds_zero_of_lt_one hr₀ hr₁).const_mul C | |
| have hlim : Tendsto (fun j : ℕ => ((j : ℝ) * K + C) * (a / b) ^ j) | |
| atTop (𝓝 0) := by | |
| convert hlinear.add hconstant using 1 <;> try simp | |
| funext j | |
| ring | |
| have hbound : ∀ᶠ j : ℕ in atTop, | |
| 1 ≤ ((j : ℝ) * K + C) * (a / b) ^ j := by | |
| filter_upwards [h] with j hj | |
| have hdiv := (div_le_div_iff_of_pos_right (pow_pos hb j)).2 hj | |
| simpa only [div_self (pow_ne_zero j hb.ne'), mul_div_assoc, div_pow] using hdiv | |
| have hfalse : (1 : ℝ) ≤ 0 := ge_of_tendsto hlim hbound | |
| linarith | |
| /-- Removing the linear interpolation overhead from inequalities for all powers. -/ | |
| theorem le_of_pow_le_linear_mul_pow {a b K : ℝ} (ha : 0 ≤ a) | |
| (h : ∀ j : ℕ, b ^ j ≤ ((j : ℝ) * K + 1) * a ^ j) : b ≤ a := | |
| le_of_eventually_pow_le_linear_mul_pow ha (Eventually.of_forall h) | |
| /-- Fixed multiplicative losses disappear on taking exponential growth rates. -/ | |
| theorem le_of_eventually_pow_le_const_mul_pow {a b C : ℝ} (ha : 0 ≤ a) | |
| (h : ∀ᶠ j : ℕ in atTop, b ^ j ≤ C * a ^ j) : b ≤ a := by | |
| apply le_of_eventually_pow_le_linear_mul_pow (K := 0) (C := C) ha | |
| simpa only [mul_zero, zero_add] using h | |
| /-- An all-powers form of constant-factor removal. -/ | |
| theorem le_of_pow_le_const_mul_pow {a b C : ℝ} (ha : 0 ≤ a) | |
| (h : ∀ j : ℕ, b ^ j ≤ C * a ^ j) : b ≤ a := | |
| le_of_eventually_pow_le_const_mul_pow ha (Eventually.of_forall h) | |
| /-- In particular, the fixed factor five in the type-counting bound is harmless. -/ | |
| theorem le_of_pow_le_five_mul_pow {a b : ℝ} (ha : 0 ≤ a) | |
| (h : ∀ j : ℕ, b ^ j ≤ 5 * a ^ j) : b ≤ a := | |
| le_of_eventually_pow_le_const_mul_pow ha (Eventually.of_forall h) | |
| /-- A fixed counting loss does not contribute to a normalized logarithmic | |
| growth rate. The exponent may run through any sequence tending to infinity, | |
| including the multiples of a common denominator used for rational types. -/ | |
| theorem exp_le_of_tendsto_log_div {u : ℕ → ℝ} {n : ℕ → ℕ} {b C L : ℝ} | |
| (hb : 0 < b) (hC : 0 < C) (hn : Tendsto n atTop atTop) | |
| (hu : ∀ᶠ k in atTop, 0 < u k) | |
| (hlim : Tendsto (fun k => Real.log (u k) / (n k : ℝ)) atTop (𝓝 L)) | |
| (hbound : ∀ᶠ k in atTop, u k ≤ C * b ^ n k) : | |
| Real.exp L ≤ b := by | |
| have hconstant : Tendsto (fun k => Real.log C / (n k : ℝ) + Real.log b) | |
| atTop (𝓝 (Real.log b)) := by | |
| simpa using ((tendsto_const_div_atTop_nhds_zero_nat (Real.log C)).comp hn).add_const | |
| (Real.log b) | |
| have hnormalized : ∀ᶠ k in atTop, | |
| Real.log (u k) / (n k : ℝ) ≤ Real.log C / (n k : ℝ) + Real.log b := by | |
| filter_upwards [hu, hbound, hn.eventually (eventually_gt_atTop 0)] with k huk hk hnk | |
| have hnpos : (0 : ℝ) < n k := by exact_mod_cast hnk | |
| have hlog := Real.log_le_log huk hk | |
| rw [Real.log_mul hC.ne' (pow_ne_zero _ hb.ne'), Real.log_pow] at hlog | |
| calc | |
| Real.log (u k) / (n k : ℝ) ≤ | |
| (Real.log C + (n k : ℝ) * Real.log b) / (n k : ℝ) := | |
| (div_le_div_iff_of_pos_right hnpos).2 hlog | |
| _ = Real.log C / (n k : ℝ) + Real.log b := by | |
| field_simp | |
| have hL : L ≤ Real.log b := le_of_tendsto_of_tendsto hlim hconstant hnormalized | |
| simpa only [Real.exp_log hb] using Real.exp_le_exp.mpr hL | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |