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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | 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
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