| import Mathlib.Algebra.Order.Archimedean.Real.Basic |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Positivity |
| import OAI.Computability.UniqueGames.Inverse.AffineWitnessLemmas |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! Logical dimension is chosen after the inverse density and row bound, |
| and before the tuple length. All row maps, including deficient-rank maps, |
| satisfy the resulting small trivial-character bound. -/ |
|
|
| namespace UniqueGamesTheorem.Decoder.DimensionSelection |
|
|
| noncomputable section |
|
|
| theorem exists_logical_dimension {α η : ℝ} (hα : 0 < α) (hη : 0 < η) |
| (r lower : ℕ) : |
| ∃ ell : ℕ, lower ≤ ell ∧ r < ell ∧ |
| 1 / (2 : ℝ) ^ (ell - r) < α / 8 ∧ |
| 1 / (2 : ℝ) ^ ell < η / 4 := by |
| obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one |
| (lt_min (show (0 : ℝ) < α / 8 by positivity) |
| (show (0 : ℝ) < η / 4 by positivity)) |
| (show (1 / 2 : ℝ) < 1 by norm_num) |
| let ell := max lower (r + n + 1) |
| have hrn : r + n + 1 ≤ ell := le_max_right _ _ |
| have hr : r < ell := lt_of_lt_of_le (by omega) hrn |
| have hnsub : n ≤ ell - r := by omega |
| have hnell : n ≤ ell := hnsub.trans (Nat.sub_le _ _) |
| have hpow (j : ℕ) (hj : n ≤ j) : |
| 1 / (2 : ℝ) ^ j < min (α / 8) (η / 4) := by |
| have hmon : (1 / 2 : ℝ) ^ j ≤ (1 / 2 : ℝ) ^ n := |
| pow_le_pow_of_le_one (by norm_num) (by norm_num) hj |
| simpa only [one_div, inv_pow] using hmon.trans_lt hn |
| exact ⟨ell, le_max_left _ _, hr, |
| (hpow (ell - r) hnsub).trans_le (min_le_left _ _), |
| (hpow ell hnell).trans_le (min_le_right _ _)⟩ |
|
|
| / |
| possible public row map without conditioning on its rank. -/ |
| theorem kernel_fraction_le {ell r : ℕ} |
| (A : (Fin ell → ZMod 2) →ₗ[ZMod 2] (Fin r → ZMod 2)) [Fintype A.ker] : |
| 1 / (Fintype.card A.ker : ℝ) ≤ 1 / (2 : ℝ) ^ (ell - r) := by |
| have hcard : (2 : ℝ) ^ (ell - r) ≤ (Fintype.card A.ker : ℝ) := by |
| have h := Inverse.binary_kernel_card_lower A |
| rw [Nat.card_eq_fintype_card] at h |
| exact_mod_cast h |
| exact one_div_le_one_div_of_le (by positivity) hcard |
|
|
| theorem kernel_fraction_lt {ell r : ℕ} {α : ℝ} |
| (A : (Fin ell → ZMod 2) →ₗ[ZMod 2] (Fin r → ZMod 2)) [Fintype A.ker] |
| (hsmall : 1 / (2 : ℝ) ^ (ell - r) < α / 8) : |
| 1 / (Fintype.card A.ker : ℝ) < α / 8 := |
| (kernel_fraction_le A).trans_lt hsmall |
|
|
| end |
| end UniqueGamesTheorem.Decoder.DimensionSelection |
|
|
| end |
|
|
| end OAI |
|
|
