openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /ProfileSlopes.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.ConcaveSlopes | |
| namespace OAI | |
| /-! | |
| # Shifted tripling and diagonal increments | |
| These are the two estimates on the limiting row slopes used in Section 6. | |
| Iteration retains the additive shift in the tripling inequality. Symmetry | |
| then expresses a diagonal step as one increment in each of two adjacent rows. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open Filter | |
| open scoped Topology | |
| /-- The indices obtained by repeatedly applying the affine tripling map. -/ | |
| def shiftedTriplingIndex (c h : ℕ) : ℕ → ℕ | |
| | 0 => h | |
| | j + 1 => 3 * shiftedTriplingIndex c h j + c | |
| /-- Starting at a positive index, affine tripling escapes to infinity. -/ | |
| theorem shiftedTriplingIndex_lower (c h : ℕ) (hh : 1 ≤ h) (j : ℕ) : | |
| h + j ≤ shiftedTriplingIndex c h j := by | |
| induction j with | |
| | zero => simp [shiftedTriplingIndex] | |
| | succ j ih => | |
| simp only [shiftedTriplingIndex] | |
| omega | |
| theorem shiftedTriplingIndex_tendsto (c h : ℕ) (hh : 1 ≤ h) : | |
| Tendsto (shiftedTriplingIndex c h) atTop atTop := by | |
| apply tendsto_atTop_mono _ tendsto_id | |
| intro j | |
| change j ≤ shiftedTriplingIndex c h j | |
| have := shiftedTriplingIndex_lower c h hh j | |
| omega | |
| /-- The exact affine solution, written without truncated natural subtraction. -/ | |
| theorem shiftedTriplingIndex_cast (c h j : ℕ) : | |
| (shiftedTriplingIndex c h j : ℝ) + (c : ℝ) / 2 = | |
| (3 : ℝ) ^ j * ((h : ℝ) + (c : ℝ) / 2) := by | |
| induction j with | |
| | zero => simp [shiftedTriplingIndex] | |
| | succ j ih => | |
| simp only [shiftedTriplingIndex, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, | |
| pow_succ] | |
| nlinarith | |
| /-- Iterating the tripling inequality multiplies the profile by powers of three. -/ | |
| theorem shiftedTripling_profile_lower {P : ℕ → ℝ} {c h : ℕ} (hh : 1 ≤ h) | |
| (htrip : ∀ n : ℕ, 1 ≤ n → 3 * P n ≤ P (3 * n + c)) (j : ℕ) : | |
| (3 : ℝ) ^ j * P h ≤ P (shiftedTriplingIndex c h j) := by | |
| induction j with | |
| | zero => simp [shiftedTriplingIndex] | |
| | succ j ih => | |
| have hj : 1 ≤ shiftedTriplingIndex c h j := by | |
| have := shiftedTriplingIndex_lower c h hh j | |
| omega | |
| have hstep := htrip (shiftedTriplingIndex c h j) hj | |
| simp only [shiftedTriplingIndex, pow_succ] | |
| nlinarith | |
| /-- A row with a limiting linear slope and shifted tripling lies below the | |
| corresponding affine multiple of that slope. -/ | |
| theorem profile_le_affine_slope {P : ℕ → ℝ} {g : ℝ} {c h : ℕ} | |
| (hh : 1 ≤ h) | |
| (htrip : ∀ n : ℕ, 1 ≤ n → 3 * P n ≤ P (3 * n + c)) | |
| (hlim : Tendsto (fun n : ℕ => P n / (n : ℝ)) atTop (𝓝 g)) : | |
| P h ≤ ((h : ℝ) + (c : ℝ) / 2) * g := by | |
| have hpow : Tendsto (fun j : ℕ => (3 : ℝ) ^ j) atTop atTop := | |
| tendsto_pow_atTop_atTop_of_one_lt (by norm_num) | |
| have hzero : Tendsto (fun j : ℕ => ((c : ℝ) / 2) / (3 : ℝ) ^ j) | |
| atTop (𝓝 0) := hpow.const_div_atTop _ | |
| have hindex : Tendsto | |
| (fun j : ℕ => (shiftedTriplingIndex c h j : ℝ) / (3 : ℝ) ^ j) | |
| atTop (𝓝 ((h : ℝ) + (c : ℝ) / 2)) := by | |
| have heq (j : ℕ) : | |
| (shiftedTriplingIndex c h j : ℝ) / (3 : ℝ) ^ j = | |
| (h : ℝ) + (c : ℝ) / 2 - ((c : ℝ) / 2) / (3 : ℝ) ^ j := by | |
| have hj := shiftedTriplingIndex_cast c h j | |
| have hp : (3 : ℝ) ^ j ≠ 0 := pow_ne_zero _ (by norm_num) | |
| field_simp | |
| nlinarith | |
| simpa only [heq, sub_zero] using tendsto_const_nhds.sub hzero | |
| have hproduct := (hlim.comp (shiftedTriplingIndex_tendsto c h hh)).mul hindex | |
| have hprofile : Tendsto | |
| (fun j : ℕ => P (shiftedTriplingIndex c h j) / (3 : ℝ) ^ j) | |
| atTop (𝓝 (g * ((h : ℝ) + (c : ℝ) / 2))) := by | |
| apply hproduct.congr' | |
| apply Eventually.of_forall | |
| intro j | |
| have hj : (shiftedTriplingIndex c h j : ℝ) ≠ 0 := by | |
| have := shiftedTriplingIndex_lower c h hh j | |
| exact_mod_cast (show shiftedTriplingIndex c h j ≠ 0 by omega) | |
| dsimp only [Function.comp_def] | |
| field_simp | |
| have hbound : ∀ j : ℕ, P h ≤ P (shiftedTriplingIndex c h j) / (3 : ℝ) ^ j := by | |
| intro j | |
| apply (le_div_iff₀ (pow_pos (by norm_num : (0 : ℝ) < 3) j)).mpr | |
| simpa only [mul_comm] using shiftedTripling_profile_lower hh htrip j | |
| simpa only [mul_comm] using ge_of_tendsto hprofile (Eventually.of_forall hbound) | |
| /-- The row estimate in the positive-index notation of the paper. -/ | |
| theorem profile_row_le_slope {P : ℕ → ℕ → ℝ} {g : ℕ → ℝ} | |
| (htrip : ∀ a : ℕ, 1 ≤ a → ∀ h : ℕ, 1 ≤ h → | |
| 3 * P a h ≤ P a (3 * h + a - 1)) | |
| (hlim : ∀ a : ℕ, 1 ≤ a → | |
| Tendsto (fun h : ℕ => P a h / (h : ℝ)) atTop (𝓝 (g a))) | |
| (a h : ℕ) (ha : 1 ≤ a) (hh : 1 ≤ h) : | |
| P a h ≤ ((h : ℝ) + ((a : ℝ) - 1) / 2) * g a := by | |
| have ht : ∀ n : ℕ, 1 ≤ n → 3 * P a n ≤ P a (3 * n + (a - 1)) := by | |
| intro n hn | |
| have hi : 3 * n + (a - 1) = 3 * n + a - 1 := by omega | |
| rw [hi] | |
| exact htrip a ha n hn | |
| have hcast : ((a - 1 : ℕ) : ℝ) = (a : ℝ) - 1 := by | |
| rw [Nat.cast_sub ha, Nat.cast_one] | |
| simpa only [hcast] using profile_le_affine_slope hh ht (hlim a ha) | |
| /-- The diagonal is bounded above by the limiting row slope with the precise | |
| coefficient `(3a-1)/2`. -/ | |
| theorem profile_diagonal_le_slope {P : ℕ → ℕ → ℝ} {g : ℕ → ℝ} | |
| (htrip : ∀ a : ℕ, 1 ≤ a → ∀ h : ℕ, 1 ≤ h → | |
| 3 * P a h ≤ P a (3 * h + a - 1)) | |
| (hlim : ∀ a : ℕ, 1 ≤ a → | |
| Tendsto (fun h : ℕ => P a h / (h : ℝ)) atTop (𝓝 (g a))) | |
| (a : ℕ) (ha : 1 ≤ a) : | |
| P a a ≤ ((3 * (a : ℝ) - 1) / 2) * g a := by | |
| have hc : (a : ℝ) + ((a : ℝ) - 1) / 2 = (3 * (a : ℝ) - 1) / 2 := by ring | |
| simpa only [hc] using profile_row_le_slope htrip hlim a a ha ha | |
| /-- Symmetry identifies a diagonal step with one increment in each adjacent | |
| row. Only positive-index symmetry and slope bounds are used. -/ | |
| theorem diagonal_increment_of_slope_lower {P : ℕ → ℕ → ℝ} {g : ℕ → ℝ} | |
| (hsymm : ∀ a : ℕ, 1 ≤ a → ∀ b : ℕ, 1 ≤ b → P a b = P b a) | |
| (hinc : ∀ a : ℕ, 1 ≤ a → ∀ h : ℕ, 1 ≤ h → | |
| g a ≤ P a (h + 1) - P a h) | |
| (n : ℕ) (hn : 1 ≤ n) : | |
| P n n + g n + g (n + 1) ≤ P (n + 1) (n + 1) := by | |
| have hfirst := hinc n hn n hn | |
| have hsecond := hinc (n + 1) (by omega) n hn | |
| have hsym := hsymm n hn (n + 1) (by omega) | |
| linarith | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |