openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /ConcaveSlopes.lean
| import Mathlib.Analysis.Asymptotics.SpecificAsymptotics | |
| import Mathlib.Topology.Order.MonotoneConvergence | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! | |
| # Slopes of positive discretely concave sequences | |
| The row sequences in Section 6 have decreasing consecutive increments. A | |
| nonnegative sequence cannot have a negative increment of this kind: every later | |
| increment would be at least as negative. Consequently the increments converge | |
| to a nonnegative limit, and the sequence divided by its index has the same limit. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open Filter | |
| open scoped Topology | |
| /-- The usual second-difference inequality makes the increments decreasing. -/ | |
| theorem antitone_increments_of_discrete_concavity {u : ℕ → ℝ} | |
| (hu : ∀ n, u (n + 2) + u n ≤ 2 * u (n + 1)) : | |
| Antitone (fun n => u (n + 1) - u n) := by | |
| apply antitone_nat_of_succ_le | |
| intro n | |
| have h := hu n | |
| simp only [Nat.add_assoc] at * | |
| linarith | |
| /-- A tangent-line upper bound for a sequence with decreasing increments. -/ | |
| theorem concave_sequence_le_affine {u : ℕ → ℝ} | |
| (hd : Antitone (fun n => u (n + 1) - u n)) (n k : ℕ) : | |
| u (n + k) ≤ u n + (k : ℝ) * (u (n + 1) - u n) := by | |
| induction k with | |
| | zero => simp | |
| | succ k ih => | |
| have hstep := hd (show n ≤ n + k by omega) | |
| simp only at hstep | |
| rw [Nat.cast_add, Nat.cast_one] | |
| have hindex : n + (k + 1) = n + k + 1 := by omega | |
| rw [hindex] | |
| nlinarith | |
| /-- Every increment of a nonnegative discretely concave sequence is nonnegative. -/ | |
| theorem increment_nonneg_of_antitone {u : ℕ → ℝ} | |
| (hu : ∀ n, 0 ≤ u n) (hd : Antitone (fun n => u (n + 1) - u n)) | |
| (n : ℕ) : 0 ≤ u (n + 1) - u n := by | |
| by_contra hn | |
| have hneg : u (n + 1) - u n < 0 := lt_of_not_ge hn | |
| obtain ⟨k, hk⟩ := exists_lt_nsmul (neg_pos.mpr hneg) (u n) | |
| have hk' : u n < (k : ℝ) * -(u (n + 1) - u n) := by | |
| simpa only [nsmul_eq_mul] using hk | |
| have hbound := concave_sequence_le_affine hd n k | |
| have hnonneg := hu (n + k) | |
| nlinarith | |
| /-- A nonnegative discretely concave sequence is increasing. -/ | |
| theorem monotone_of_nonneg_of_antitone_increments {u : ℕ → ℝ} | |
| (hu : ∀ n, 0 ≤ u n) (hd : Antitone (fun n => u (n + 1) - u n)) : | |
| Monotone u := by | |
| apply monotone_nat_of_le_succ | |
| intro n | |
| have := increment_nonneg_of_antitone hu hd n | |
| linarith | |
| /-- Telescoping and Cesàro convergence identify the linear growth rate. -/ | |
| theorem tendsto_div_nat_of_tendsto_increments {u : ℕ → ℝ} {g : ℝ} | |
| (hd : Tendsto (fun n => u (n + 1) - u n) atTop (𝓝 g)) : | |
| Tendsto (fun n => u n / (n : ℝ)) atTop (𝓝 g) := by | |
| have hmean := hd.cesaro | |
| simp only [Finset.sum_range_sub] at hmean | |
| have hconst := tendsto_const_div_atTop_nhds_zero_nat (u 0) | |
| have hsum := hmean.add hconst | |
| simpa only [add_zero, div_eq_mul_inv, mul_sub, sub_add_cancel, mul_comm] using hsum | |
| /-- The limiting increment of a nonnegative discretely concave sequence is | |
| nonnegative, and it is also its linear growth rate. -/ | |
| theorem exists_nonneg_limit_of_antitone_increments {u : ℕ → ℝ} | |
| (hu : ∀ n, 0 ≤ u n) (hd : Antitone (fun n => u (n + 1) - u n)) : | |
| ∃ g : ℝ, 0 ≤ g ∧ | |
| Tendsto (fun n => u (n + 1) - u n) atTop (𝓝 g) ∧ | |
| Tendsto (fun n => u n / (n : ℝ)) atTop (𝓝 g) := by | |
| have hnonneg := increment_nonneg_of_antitone hu hd | |
| have hbdd : BddBelow (Set.range (fun n => u (n + 1) - u n)) := by | |
| refine ⟨0, ?_⟩ | |
| rintro _ ⟨n, rfl⟩ | |
| exact hnonneg n | |
| have hlim := tendsto_atTop_ciInf hd hbdd | |
| refine ⟨⨅ n, u (n + 1) - u n, ?_, hlim, | |
| tendsto_div_nat_of_tendsto_increments hlim⟩ | |
| exact ge_of_tendsto hlim (Eventually.of_forall hnonneg) | |
| /-- Positive-index version, suitable for the rows in Section 6. The value at | |
| index zero is irrelevant. -/ | |
| theorem exists_concave_slope_limit {P : ℕ → ℝ} | |
| (hP : ∀ n, 0 < P (n + 1)) | |
| (hd : Antitone (fun n => P (n + 2) - P (n + 1))) : | |
| ∃ g : ℝ, 0 ≤ g ∧ | |
| (∀ n, g ≤ P (n + 2) - P (n + 1)) ∧ | |
| Tendsto (fun n => P (n + 2) - P (n + 1)) atTop (𝓝 g) ∧ | |
| Tendsto (fun n => P n / (n : ℝ)) atTop (𝓝 g) := by | |
| have hd' : Antitone (fun n => P ((n + 1) + 1) - P (n + 1)) := by | |
| simpa only [Nat.add_assoc] using hd | |
| obtain ⟨g, hg, hlim, hratio⟩ := exists_nonneg_limit_of_antitone_increments | |
| (u := fun n => P (n + 1)) (fun n => (hP n).le) hd' | |
| have hmul := hratio.mul (tendsto_natCast_div_add_atTop (1 : ℝ)) | |
| have hshift : Tendsto (fun n => P (n + 1) / ((n + 1 : ℕ) : ℝ)) | |
| atTop (𝓝 g) := by | |
| rw [mul_one] at hmul | |
| apply hmul.congr' | |
| filter_upwards [eventually_ge_atTop 1] with n hn | |
| have hn' : (n : ℝ) ≠ 0 := by exact_mod_cast (show n ≠ 0 by omega) | |
| rw [Nat.cast_add, Nat.cast_one] | |
| field_simp | |
| refine ⟨g, hg, ?_, ?_, (tendsto_add_atTop_iff_nat 1).mp hshift⟩ | |
| · intro n | |
| exact hd.le_of_tendsto (by simpa only [Nat.add_assoc] using hlim) n | |
| · simpa only [Nat.add_assoc] using hlim | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |