openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Algebra /MatrixInverseConstantsLemmas.lean
| import Mathlib.Algebra.Order.Archimedean.Basic | |
| import OAI.Computability.PerfectCompleteness.Foundations.GlobalityLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.LinearEvaluationLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.MatrixInverseConstants | |
| open UniqueGamesTheorem.Appendix UniqueGamesTheorem.Fourier UniqueGamesTheorem.Integration | |
| theorem exists_cutoff_density {η : ℝ} (hη : 0 < η) : | |
| ∃ r : ℕ, 1 ≤ r ∧ ∃ ρ : ℝ, 0 < ρ ∧ ρ < 1 ∧ | |
| MatrixLevelBridge.levelCutoffConstant r ρ + | |
| RankLevelFilter.node (r + 1) < η := by | |
| obtain ⟨q, hq, hqη⟩ := exists_pos_rat_lt hη | |
| obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one | |
| (show 0 < η / 2 by positivity) | |
| (by norm_num : (1 / 2 : ℝ) < 1) | |
| let r : ℕ := n + 1 | |
| have hn' : RankLevelFilter.node n < η / 2 := by | |
| simpa only [RankLevelFilter.node_eq_inv_pow, one_div, inv_pow] using hn | |
| have htail : RankLevelFilter.node (r + 1) < η / 2 := | |
| (RankLevelFilter.node_antitone (show n ≤ r + 1 by dsimp [r]; omega)).trans_lt hn' | |
| obtain ⟨s, hs, hcut⟩ := CutoffBudget.exists_symbol_dimension_cutoff r hq | |
| have hρ := CutoffBudget.rho_real_bounds hs | |
| refine ⟨r, by dsimp [r]; omega, (FinalParameters.rho r s : ℝ), hρ.1, hρ.2, ?_⟩ | |
| have hcut' : | |
| MatrixLevelBridge.levelCutoffConstant r (FinalParameters.rho r s : ℝ) ≤ | |
| (q : ℝ) / 8 := by | |
| simpa only [MatrixLevelBridge.levelCutoffConstant, CutoffBudget.cutoffSum] using hcut | |
| linarith | |
| end PerfectCompleteness.MatrixInverseConstants | |
| end | |
| section | |
| namespace PerfectCompleteness.LowerRankParameters | |
| open UniqueGamesTheorem.Appendix UniqueGamesTheorem.Fourier | |
| theorem power_ratio_le {a b k : Nat} (h : a + k ≤ b) : | |
| (2 : ℝ) ^ a / (2 : ℝ) ^ b ≤ (1 / 2 : ℝ) ^ k := by | |
| rw [div_pow, one_pow] | |
| apply (div_le_div_iff₀ (by positivity) (by positivity)).2 | |
| simpa only [one_mul, ← pow_add] using | |
| (pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) h) | |
| theorem exists_cross_parameters (r : Nat) {ρ : ℝ} (hρ : 0 < ρ) : | |
| ∃ m s : Nat, 0 < m ∧ 1 ≤ s ∧ ∀ s' : Nat, s ≤ s' → | |
| (2 : ℝ) ^ m / (2 : ℝ) ^ (s' - 2 * r) + | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) < ρ := by | |
| obtain ⟨k, hk⟩ := exists_pow_lt_of_lt_one | |
| (show 0 < ρ / 2 by positivity) (by norm_num : (1 / 2 : ℝ) < 1) | |
| let m := k + 3 | |
| let s := 2 * r + m + k + 1 | |
| refine ⟨m, s, by dsimp [m]; omega, by dsimp [s]; omega, ?_⟩ | |
| intro s' hs' | |
| have hfirst : m + k ≤ s' - 2 * r := by dsimp [s] at hs'; omega | |
| have hsecond : 2 * m + k ≤ m * m := by dsimp [m]; nlinarith | |
| have h₁ := power_ratio_le hfirst | |
| have h₂ := power_ratio_le hsecond | |
| linarith | |
| theorem exists_lower_parameters {η : ℝ} (hη : 0 < η) : | |
| ∃ r m s rowMinimum : Nat, 1 ≤ r ∧ 0 < m ∧ 1 ≤ s ∧ | |
| ∃ ρ : ℝ, 0 < ρ ∧ ρ < 1 ∧ | |
| MatrixLevelBridge.levelCutoffConstant r ρ + RankLevelFilter.node (r + 1) < η ∧ | |
| (∀ ℓ, rowMinimum ≤ ℓ → 2 * m + r ≤ ℓ) ∧ | |
| ∀ s', s ≤ s' → | |
| (2 : ℝ) ^ m / (2 : ℝ) ^ (s' - 2 * r) + | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) < ρ := by | |
| obtain ⟨r, hr, ρ, hρ, hρ', hcut⟩ := MatrixInverseConstants.exists_cutoff_density hη | |
| obtain ⟨m, s, hm, hs, herror⟩ := exists_cross_parameters r hρ | |
| exact ⟨r, m, s, 2 * m + r, hr, hm, hs, ρ, hρ, hρ', hcut, | |
| fun _ h => h, herror⟩ | |
| end PerfectCompleteness.LowerRankParameters | |
| end | |
| end OAI | |