openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /MultiplicativePower.lean
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| import Mathlib.Algebra.Order.Archimedean.Basic | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! | |
| # Monotone multiplicative functions on the positive natural numbers | |
| The scalar classification used in Section 3.1: positivity, multiplicativity, | |
| and monotonicity force a function on positive natural numbers to be a real | |
| power. No property of the function at zero is required. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- Multiplicativity and positivity determine the value at one. -/ | |
| theorem positiveMultiplicative_one | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) : | |
| f 1 = 1 := by | |
| have hm := hmul 1 1 (by omega) (by omega) | |
| have hp := hpos 1 (by omega) | |
| norm_num at hm | |
| nlinarith | |
| /-- A positive multiplicative function preserves natural powers. -/ | |
| theorem positiveMultiplicative_pow | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| {n : ℕ} (hn : 0 < n) (k : ℕ) : f (n ^ k) = f n ^ k := by | |
| induction k with | |
| | zero => simpa using positiveMultiplicative_one hpos hmul | |
| | succ k ih => | |
| rw [pow_succ, hmul (n ^ k) n (pow_pos hn k) hn, ih, pow_succ] | |
| /-- Monotonicity transfers comparisons of natural powers to the values. -/ | |
| theorem positiveMultiplicative_pow_le_pow | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| (hmono : MonotoneOn f (Set.Ioi 0)) | |
| {m n : ℕ} (hm : 0 < m) (hn : 0 < n) {j k : ℕ} | |
| (h : m ^ j ≤ n ^ k) : f m ^ j ≤ f n ^ k := by | |
| rw [← positiveMultiplicative_pow hpos hmul hm, | |
| ← positiveMultiplicative_pow hpos hmul hn] | |
| exact hmono (pow_pos hm j) (pow_pos hn k) h | |
| /-- The same comparison after taking logarithms. -/ | |
| theorem positiveMultiplicative_log_pow_le | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| (hmono : MonotoneOn f (Set.Ioi 0)) | |
| {m n : ℕ} (hm : 0 < m) (hn : 0 < n) {j k : ℕ} | |
| (h : m ^ j ≤ n ^ k) : | |
| (j : ℝ) * Real.log (f m) ≤ (k : ℝ) * Real.log (f n) := by | |
| have hv := positiveMultiplicative_pow_le_pow hpos hmul hmono hm hn h | |
| simpa only [Real.log_pow] using | |
| Real.log_le_log (pow_pos (hpos m hm) j) hv | |
| /-- Bounds from neighboring powers of two have a fixed logarithmic error, | |
| independent of the exponent `k`. -/ | |
| theorem positiveMultiplicative_log_error_bounds | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| (hmono : MonotoneOn f (Set.Ioi 0)) | |
| {n : ℕ} (hn : 0 < n) (k : ℕ) : | |
| (k : ℝ) * (Real.log (f n) * Real.log 2 - | |
| Real.log (n : ℝ) * Real.log (f 2)) ≤ Real.log (f 2) * Real.log 2 ∧ | |
| (k : ℝ) * (Real.log (n : ℝ) * Real.log (f 2) - | |
| Real.log (f n) * Real.log 2) ≤ Real.log (f 2) * Real.log 2 := by | |
| have hf2 : 0 ≤ Real.log (f 2) := by | |
| apply Real.log_nonneg | |
| have hm := hmono (show (1 : ℕ) ∈ Set.Ioi 0 by norm_num) | |
| (show (2 : ℕ) ∈ Set.Ioi 0 by norm_num) (by norm_num) | |
| rwa [positiveMultiplicative_one hpos hmul] at hm | |
| have hlog2 : 0 ≤ Real.log (2 : ℝ) := Real.log_nonneg (by norm_num) | |
| obtain ⟨m, hlow, hupp⟩ := | |
| exists_nat_pow_near (show 1 ≤ n ^ k from Nat.one_le_iff_ne_zero.mpr (by positivity)) | |
| (show 1 < (2 : ℕ) by omega) | |
| have hflow := positiveMultiplicative_log_pow_le hpos hmul hmono | |
| (show 0 < (2 : ℕ) by omega) hn hlow | |
| have hfupp := positiveMultiplicative_log_pow_le hpos hmul hmono hn | |
| (show 0 < (2 : ℕ) by omega) hupp.le | |
| have hlow' : (2 : ℝ) ^ m ≤ (n : ℝ) ^ k := by exact_mod_cast hlow | |
| have hupp' : (n : ℝ) ^ k ≤ (2 : ℝ) ^ (m + 1) := by exact_mod_cast hupp.le | |
| have hnp : (0 : ℝ) < n := by exact_mod_cast hn | |
| have hnlow := Real.log_le_log (pow_pos (by norm_num : (0 : ℝ) < 2) m) hlow' | |
| have hnupp := Real.log_le_log (pow_pos hnp k) hupp' | |
| simp only [Real.log_pow, Nat.cast_add, Nat.cast_one] at hflow hfupp hnlow hnupp | |
| have h₁ := mul_le_mul_of_nonneg_right hfupp hlog2 | |
| have h₂ := mul_le_mul_of_nonneg_right hnlow hf2 | |
| have h₃ := mul_le_mul_of_nonneg_right hnupp hf2 | |
| have h₄ := mul_le_mul_of_nonneg_right hflow hlog2 | |
| constructor <;> nlinarith | |
| /-- If every natural multiple of both differences has a common upper bound, | |
| the two real numbers coincide. -/ | |
| theorem eq_of_nat_mul_sub_bounded {x y C : ℝ} | |
| (hxy : ∀ k : ℕ, (k : ℝ) * (x - y) ≤ C) | |
| (hyx : ∀ k : ℕ, (k : ℝ) * (y - x) ≤ C) : x = y := by | |
| apply le_antisymm | |
| · by_contra h | |
| have hd : 0 < x - y := sub_pos.mpr (lt_of_not_ge h) | |
| obtain ⟨k, hk⟩ := exists_nat_gt (C / (x - y)) | |
| have hk' : C < (k : ℝ) * (x - y) := (div_lt_iff₀ hd).mp hk | |
| exact (not_lt_of_ge (hxy k)) hk' | |
| · by_contra h | |
| have hd : 0 < y - x := sub_pos.mpr (lt_of_not_ge h) | |
| obtain ⟨k, hk⟩ := exists_nat_gt (C / (y - x)) | |
| have hk' : C < (k : ℝ) * (y - x) := (div_lt_iff₀ hd).mp hk | |
| exact (not_lt_of_ge (hyx k)) hk' | |
| /-- The logarithmic exponent is the same at every positive natural number. -/ | |
| theorem positiveMultiplicative_log_eq | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| (hmono : MonotoneOn f (Set.Ioi 0)) | |
| {n : ℕ} (hn : 0 < n) : | |
| Real.log (f n) = Real.log (n : ℝ) * (Real.log (f 2) / Real.log 2) := by | |
| have heq : Real.log (f n) * Real.log 2 = | |
| Real.log (n : ℝ) * Real.log (f 2) := | |
| eq_of_nat_mul_sub_bounded | |
| (fun k => (positiveMultiplicative_log_error_bounds hpos hmul hmono hn k).1) | |
| (fun k => (positiveMultiplicative_log_error_bounds hpos hmul hmono hn k).2) | |
| calc | |
| Real.log (f n) = (Real.log (n : ℝ) * Real.log (f 2)) / Real.log 2 := | |
| (eq_div_iff (Real.log_pos (show (1 : ℝ) < 2 by norm_num)).ne').2 heq | |
| _ = Real.log (n : ℝ) * (Real.log (f 2) / Real.log 2) := by ring | |
| /-- The explicit exponent in the classification of positive monotone | |
| multiplicative functions on the positive natural numbers. -/ | |
| theorem positiveMultiplicative_eq_rpow | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| (hmono : MonotoneOn f (Set.Ioi 0)) | |
| {n : ℕ} (hn : 0 < n) : | |
| f n = (n : ℝ) ^ (Real.log (f 2) / Real.log 2) := by | |
| have hnp : (0 : ℝ) < n := by exact_mod_cast hn | |
| rw [Real.rpow_def_of_pos hnp, ← positiveMultiplicative_log_eq hpos hmul hmono hn, | |
| Real.exp_log (hpos n hn)] | |
| /-- Every positive monotone multiplicative function on the positive natural | |
| numbers is a nonnegative real power, as used in Section 3.1. -/ | |
| theorem positiveMultiplicative_is_rpow | |
| {f : ℕ → ℝ} (hpos : ∀ n, 0 < n → 0 < f n) | |
| (hmul : ∀ m n, 0 < m → 0 < n → f (m * n) = f m * f n) | |
| (hmono : MonotoneOn f (Set.Ioi 0)) : | |
| ∃ p : ℝ, 0 ≤ p ∧ ∀ n : ℕ, 0 < n → f n = (n : ℝ) ^ p := by | |
| refine ⟨Real.log (f 2) / Real.log 2, ?_, fun n hn => | |
| positiveMultiplicative_eq_rpow hpos hmul hmono hn⟩ | |
| apply div_nonneg _ (Real.log_nonneg (by norm_num)) | |
| apply Real.log_nonneg | |
| have hm := hmono (show (1 : ℕ) ∈ Set.Ioi 0 by norm_num) | |
| (show (2 : ℕ) ∈ Set.Ioi 0 by norm_num) (by norm_num) | |
| rwa [positiveMultiplicative_one hpos hmul] at hm | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |