openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /Recurrence.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.Product | |
| namespace OAI | |
| /-! | |
| # The scalar simultaneous induction in Section 6 | |
| The two diagonal inequalities imply, simultaneously, lower bounds for the | |
| limiting row slopes and for the diagonal profile. All profile properties | |
| needed for this argument are expressed as hypotheses below. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- One step of the Section 6 simultaneous induction, with a positive real | |
| index. The proof uses the diagonal increment inequality to obtain the second | |
| conclusion as well as the first. -/ | |
| theorem growthRecurrence_step_real | |
| {n Dprev Dnext gprev gnext Gprev : ℝ} (hn : 0 < n) | |
| (hupper : Dnext ≤ ((3 * (n + 1) - 1) / 2) * gnext) | |
| (hincrement : Dprev + gprev + gnext ≤ Dnext) | |
| (hgprev : Gprev ≤ gprev) | |
| (hDprev : ((3 * n - 1) / 2) * Gprev ≤ Dprev) : | |
| (1 + 1 / (3 * n)) * Gprev ≤ gnext ∧ | |
| ((3 * (n + 1) - 1) / 2) * ((1 + 1 / (3 * n)) * Gprev) ≤ Dnext := by | |
| have hden : 0 < 3 * n := by positivity | |
| have hbound : (3 * n + 1) * Gprev ≤ (3 * n) * gnext := by | |
| nlinarith | |
| have hidentity : (1 + 1 / (3 * n)) * Gprev = | |
| ((3 * n + 1) * Gprev) / (3 * n) := by | |
| field_simp | |
| have hgnew : (1 + 1 / (3 * n)) * Gprev ≤ gnext := by | |
| rw [hidentity] | |
| exact (div_le_iff₀ hden).mpr (by simpa only [mul_comm] using hbound) | |
| refine ⟨hgnew, ?_⟩ | |
| have hdiagonal : ((3 * (n + 1) - 1) / 2) * | |
| ((1 + 1 / (3 * n)) * Gprev) = | |
| ((3 * n - 1) / 2) * Gprev + Gprev + | |
| (1 + 1 / (3 * n)) * Gprev := by | |
| field_simp | |
| ring | |
| rw [hdiagonal] | |
| linarith | |
| /-- The natural-number form of the scalar induction step, indexed by `n + 1` | |
| so that no truncated subtraction occurs in the recurrence. -/ | |
| theorem growthRecurrence_step | |
| {n : ℕ} (hn : 1 ≤ n) {Dprev Dnext gprev gnext Gprev : ℝ} | |
| (hupper : Dnext ≤ ((3 * ((n : ℝ) + 1) - 1) / 2) * gnext) | |
| (hincrement : Dprev + gprev + gnext ≤ Dnext) | |
| (hgprev : Gprev ≤ gprev) | |
| (hDprev : ((3 * (n : ℝ) - 1) / 2) * Gprev ≤ Dprev) : | |
| (1 + 1 / (3 * (n : ℝ))) * Gprev ≤ gnext ∧ | |
| ((3 * ((n : ℝ) + 1) - 1) / 2) * | |
| ((1 + 1 / (3 * (n : ℝ))) * Gprev) ≤ Dnext := by | |
| apply growthRecurrence_step_real | |
| · exact_mod_cast (show 0 < n by omega) | |
| · exact hupper | |
| · exact hincrement | |
| · exact hgprev | |
| · exact hDprev | |
| /-- The simultaneous slope and diagonal estimates, conditional only on the | |
| initial values, the recurrence for `G`, and the two diagonal inequalities. -/ | |
| theorem growthRecurrence_induction | |
| (D g G : ℕ → ℝ) | |
| (hGbase : G 1 = 1) | |
| (hgbase : 1 ≤ g 1) | |
| (hDbase : 1 ≤ D 1) | |
| (hGstep : ∀ n : ℕ, 1 ≤ n → | |
| G (n + 1) = (1 + 1 / (3 * (n : ℝ))) * G n) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → | |
| D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n) | |
| (hincrement : ∀ n : ℕ, 1 ≤ n → | |
| D n + g n + g (n + 1) ≤ D (n + 1)) : | |
| ∀ n : ℕ, 1 ≤ n → G n ≤ g n ∧ ((3 * (n : ℝ) - 1) / 2) * G n ≤ D n := by | |
| intro n hn | |
| induction n, hn using Nat.le_induction with | |
| | base => | |
| norm_num [hGbase] | |
| exact ⟨hgbase, hDbase⟩ | |
| | succ n hn ih => | |
| have hu : D (n + 1) ≤ | |
| ((3 * ((n : ℝ) + 1) - 1) / 2) * g (n + 1) := by | |
| simpa only [Nat.cast_add, Nat.cast_one] using hupper (n + 1) (by omega) | |
| have hs := growthRecurrence_step hn hu (hincrement n hn) ih.1 ih.2 | |
| simpa only [hGstep n hn, Nat.cast_add, Nat.cast_one] using hs | |
| /-- A zero-indexed version of `growthRecurrence_induction`, convenient when the | |
| initial term of the comparison sequence is stored at index zero. -/ | |
| theorem growthRecurrence_induction_shifted | |
| (D g H : ℕ → ℝ) | |
| (hHbase : H 0 = 1) | |
| (hgbase : 1 ≤ g 1) | |
| (hDbase : 1 ≤ D 1) | |
| (hHstep : ∀ n : ℕ, | |
| H (n + 1) = (1 + 1 / (3 * ((n : ℝ) + 1))) * H n) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → | |
| D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n) | |
| (hincrement : ∀ n : ℕ, 1 ≤ n → | |
| D n + g n + g (n + 1) ≤ D (n + 1)) : | |
| ∀ n : ℕ, H n ≤ g (n + 1) ∧ | |
| ((3 * ((n : ℝ) + 1) - 1) / 2) * H n ≤ D (n + 1) := by | |
| intro n | |
| induction n with | |
| | zero => | |
| norm_num [hHbase] | |
| exact ⟨hgbase, hDbase⟩ | |
| | succ n ih => | |
| have hn : 1 ≤ n + 1 := by omega | |
| have hu : D (n + 1 + 1) ≤ | |
| ((3 * (((n + 1 : ℕ) : ℝ) + 1) - 1) / 2) * g (n + 1 + 1) := by | |
| simpa only [Nat.cast_add, Nat.cast_one] using | |
| hupper (n + 1 + 1) (by omega) | |
| have hp : ((3 * ((n + 1 : ℕ) : ℝ) - 1) / 2) * H n ≤ D (n + 1) := by | |
| simpa only [Nat.cast_add, Nat.cast_one] using ih.2 | |
| have hs := growthRecurrence_step hn hu (hincrement (n + 1) hn) ih.1 hp | |
| simpa only [hHstep, Nat.cast_add, Nat.cast_one] using hs | |
| /-- The simultaneous bounds for the explicit comparison product from Section 6. -/ | |
| theorem growthProduct_le_slope_and_diagonal | |
| (D g : ℕ → ℝ) | |
| (hgbase : 1 ≤ g 1) | |
| (hDbase : 1 ≤ D 1) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → | |
| D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n) | |
| (hincrement : ∀ n : ℕ, 1 ≤ n → | |
| D n + g n + g (n + 1) ≤ D (n + 1)) : | |
| ∀ n : ℕ, growthProduct n ≤ g (n + 1) ∧ | |
| ((3 * ((n : ℝ) + 1) - 1) / 2) * growthProduct n ≤ D (n + 1) := by | |
| apply growthRecurrence_induction_shifted D g growthProduct | |
| growthProduct_zero hgbase hDbase | |
| · intro n | |
| rw [growthProduct_succ, mul_comm] | |
| · exact hupper | |
| · exact hincrement | |
| /-- Combining the scalar induction with the elementary product estimate gives | |
| an exact fourth-power lower bound on the cube of each diagonal value. -/ | |
| theorem diagonal_fourth_power_lower | |
| (D g : ℕ → ℝ) | |
| (hgbase : 1 ≤ g 1) | |
| (hDbase : 1 ≤ D 1) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → | |
| D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n) | |
| (hincrement : ∀ n : ℕ, 1 ≤ n → | |
| D n + g n + g (n + 1) ≤ D (n + 1)) | |
| (n : ℕ) : | |
| ((n : ℝ) + 1) ^ 4 ≤ D (n + 1) ^ 3 := by | |
| have hdiag := (growthProduct_le_slope_and_diagonal D g hgbase hDbase | |
| hupper hincrement n).2 | |
| have hpositive := growthProduct_pos n | |
| have hlinear : ((n : ℝ) + 1) * growthProduct n ≤ D (n + 1) := by | |
| have hcoeff : (n : ℝ) + 1 ≤ (3 * ((n : ℝ) + 1) - 1) / 2 := by | |
| have hn : (0 : ℝ) ≤ n := Nat.cast_nonneg n | |
| linarith | |
| exact (mul_le_mul_of_nonneg_right hcoeff (le_of_lt hpositive)).trans hdiag | |
| calc | |
| ((n : ℝ) + 1) ^ 4 = ((n : ℝ) + 1) ^ 3 * ((n : ℝ) + 1) := by ring | |
| _ ≤ ((n : ℝ) + 1) ^ 3 * growthProduct n ^ 3 := | |
| mul_le_mul_of_nonneg_left (growthProduct_cube_lower n) (by positivity) | |
| _ = (((n : ℝ) + 1) * growthProduct n) ^ 3 := by ring | |
| _ ≤ D (n + 1) ^ 3 := pow_le_pow_left₀ (by positivity) hlinear 3 | |
| /-- The diagonal fourth-power lower bound using the paper's positive indices. -/ | |
| theorem diagonal_fourth_power_lower_of_one_le | |
| (D g : ℕ → ℝ) | |
| (hgbase : 1 ≤ g 1) | |
| (hDbase : 1 ≤ D 1) | |
| (hupper : ∀ n : ℕ, 1 ≤ n → | |
| D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n) | |
| (hincrement : ∀ n : ℕ, 1 ≤ n → | |
| D n + g n + g (n + 1) ≤ D (n + 1)) | |
| (n : ℕ) (hn : 1 ≤ n) : (n : ℝ) ^ 4 ≤ D n ^ 3 := by | |
| cases n with | |
| | zero => omega | |
| | succ n => | |
| simpa only [Nat.cast_add, Nat.cast_one] using | |
| diagonal_fourth_power_lower D g hgbase hDbase hupper hincrement n | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |