openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Growth /Profile.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.ConcaveSlopes | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.ProfileSlopes | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.Recurrence | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.ExponentComparison | |
| namespace OAI | |
| /-! | |
| # Growth of the scalar profile | |
| This file proves the scalar implication of Section 6 (Lemma 6.1). The profile | |
| properties are explicit hypotheses: the construction of a profile from a tensor | |
| character and the proof of those properties are separate arguments. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open Filter | |
| open scoped Topology | |
| /-- The scalar profile hypotheses used in Section 6. Only positive indices | |
| carry mathematical content; the values on either coordinate axis are unused. -/ | |
| structure ScalarProfile (t : ℝ) where | |
| value : ℕ → ℕ → ℝ | |
| positive : ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → 0 < value a b | |
| symmetric : ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → value a b = value b a | |
| boundary : ∀ b : ℕ, 1 ≤ b → value 1 b = (b : ℝ) | |
| concave : ∀ a h : ℕ, 1 ≤ a → 2 ≤ h → | |
| value a (h - 1) + value a (h + 1) ≤ 2 * value a h | |
| tripling : ∀ a h : ℕ, 1 ≤ a → 1 ≤ h → | |
| 3 * value a h ≤ value a (3 * h + a - 1) | |
| rank_bound : ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → | |
| value a b ≤ ((a : ℝ) + (b : ℝ) - 1) ^ (1 / t) | |
| namespace ScalarProfile | |
| /-- Each row has a nonnegative limiting slope that bounds every increment | |
| from below and equals the limit of the row divided by its index. -/ | |
| theorem exists_row_slope {t : ℝ} (P : ScalarProfile t) (a : ℕ) (ha : 1 ≤ a) : | |
| ∃ g : ℝ, 0 ≤ g ∧ | |
| (∀ h : ℕ, 1 ≤ h → g ≤ P.value a (h + 1) - P.value a h) ∧ | |
| Tendsto (fun h : ℕ => P.value a h / (h : ℝ)) atTop (𝓝 g) := by | |
| have hd : Antitone (fun n : ℕ => P.value a (n + 2) - P.value a (n + 1)) := by | |
| apply antitone_nat_of_succ_le | |
| intro n | |
| have hc := P.concave a (n + 2) ha (by omega) | |
| have hpred : n + 2 - 1 = n + 1 := by omega | |
| rw [hpred] at hc | |
| simp only [Nat.add_assoc] at * | |
| linarith | |
| obtain ⟨g, hg, hinc, _, hratio⟩ := exists_concave_slope_limit | |
| (P := P.value a) (fun n => P.positive a (n + 1) ha (by omega)) hd | |
| refine ⟨g, hg, ?_, hratio⟩ | |
| intro h hh | |
| cases h with | |
| | zero => omega | |
| | succ n => simpa only [Nat.add_assoc] using hinc n | |
| /-- Choose the row slopes together, retaining their bounds and limits on all | |
| positive rows. -/ | |
| theorem exists_row_slopes {t : ℝ} (P : ScalarProfile t) : | |
| ∃ g : ℕ → ℝ, ∀ a : ℕ, 1 ≤ a → 0 ≤ g a ∧ | |
| (∀ h : ℕ, 1 ≤ h → g a ≤ P.value a (h + 1) - P.value a h) ∧ | |
| Tendsto (fun h : ℕ => P.value a h / (h : ℝ)) atTop (𝓝 (g a)) := by | |
| classical | |
| have hex (a : ℕ) : ∃ g : ℝ, 1 ≤ a → 0 ≤ g ∧ | |
| (∀ h : ℕ, 1 ≤ h → g ≤ P.value a (h + 1) - P.value a h) ∧ | |
| Tendsto (fun h : ℕ => P.value a h / (h : ℝ)) atTop (𝓝 g) := by | |
| by_cases ha : 1 ≤ a | |
| · obtain ⟨g, hg⟩ := P.exists_row_slope a ha | |
| exact ⟨g, fun _ => hg⟩ | |
| · exact ⟨0, fun ha' => False.elim (ha ha')⟩ | |
| choose g hg using hex | |
| exact ⟨g, hg⟩ | |
| /-- The exact polynomial lower bound on the diagonal, obtained from row | |
| concavity, shifted tripling, and the simultaneous scalar induction. -/ | |
| theorem diagonal_fourth_power_lower {t : ℝ} (P : ScalarProfile t) | |
| (n : ℕ) (hn : 1 ≤ n) : (n : ℝ) ^ 4 ≤ P.value n n ^ 3 := by | |
| obtain ⟨g, hg⟩ := P.exists_row_slopes | |
| have hupper : ∀ a : ℕ, 1 ≤ a → | |
| P.value a a ≤ ((3 * (a : ℝ) - 1) / 2) * g a := by | |
| apply profile_diagonal_le_slope | |
| · intro a ha h hh | |
| exact P.tripling a h ha hh | |
| · intro a ha | |
| exact (hg a ha).2.2 | |
| have hincrement : ∀ a : ℕ, 1 ≤ a → | |
| P.value a a + g a + g (a + 1) ≤ P.value (a + 1) (a + 1) := by | |
| apply diagonal_increment_of_slope_lower | |
| · intro a ha b hb | |
| exact P.symmetric a b ha hb | |
| · intro a ha h hh | |
| exact (hg a ha).2.1 h hh | |
| have hDbase : 1 ≤ P.value 1 1 := by | |
| rw [P.boundary 1 (by omega)] | |
| norm_num | |
| have hgbase : 1 ≤ g 1 := by | |
| have hu := hupper 1 (by omega) | |
| rw [P.boundary 1 (by omega)] at hu | |
| norm_num at hu | |
| exact hu | |
| exact MatrixMultiplication.AuxiliarySeparation.diagonal_fourth_power_lower_of_one_le | |
| (fun a => P.value a a) g hgbase hDbase hupper hincrement n hn | |
| /-- Lemma 6.1: a positive symmetric profile satisfying the stated concavity, | |
| tripling, boundary, and rank bounds has exponent at most `3/4`. -/ | |
| theorem exponent_le_three_quarters {t : ℝ} (P : ScalarProfile t) (ht : 0 < t) : | |
| t ≤ 3 / 4 := by | |
| apply diagonal_exponent_le_three_quarters (D := fun a => P.value a a) ht | |
| · intro n hn | |
| exact (P.positive n n hn hn).le | |
| · exact P.diagonal_fourth_power_lower | |
| · intro n hn | |
| have hu := P.rank_bound n n hn hn | |
| simpa only [two_mul] using hu | |
| end ScalarProfile | |
| /-- The scalar profile implication, with every profile assumption displayed | |
| explicitly rather than bundled in `ScalarProfile`. -/ | |
| theorem profile_exponent_le_three_quarters {P : ℕ → ℕ → ℝ} {t : ℝ} | |
| (ht : 0 < t) | |
| (hpositive : ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → 0 < P a b) | |
| (hsymmetric : ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → P a b = P b a) | |
| (hboundary : ∀ b : ℕ, 1 ≤ b → P 1 b = (b : ℝ)) | |
| (hconcave : ∀ a h : ℕ, 1 ≤ a → 2 ≤ h → | |
| P a (h - 1) + P a (h + 1) ≤ 2 * P a h) | |
| (htripling : ∀ a h : ℕ, 1 ≤ a → 1 ≤ h → | |
| 3 * P a h ≤ P a (3 * h + a - 1)) | |
| (hrank : ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → | |
| P a b ≤ ((a : ℝ) + (b : ℝ) - 1) ^ (1 / t)) : | |
| t ≤ 3 / 4 := by | |
| exact ScalarProfile.exponent_le_three_quarters | |
| ⟨P, hpositive, hsymmetric, hboundary, hconcave, htripling, hrank⟩ ht | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |