| import Mathlib.Algebra.Order.Archimedean.Basic |
| import Mathlib.Tactic.Linarith |
| import OAI.Computability.UniqueGames.Quadratic.BlockSpanLemmas |
| import OAI.Computability.UniqueGames.Quadratic.GenericSubspaceProbability |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| |
|
|
| The extension degree is chosen using only the rank threshold and the requested |
| genericity error. In particular it is fixed before the logical dimension in |
| the later enlargement construction. A generic top-rank space exists in the |
| same field. |
| -/ |
|
|
| namespace UniqueGamesTheorem.Quadratic |
|
|
| open scoped BigOperators |
| open UniqueGamesTheorem.Gadget.Orientation |
|
|
| noncomputable section |
|
|
| noncomputable instance binaryFieldFintype (d : ℕ) : Fintype (BinaryField d) := |
| Fintype.ofFinite _ |
|
|
| / |
| used by the latent gadget. The conclusion covers rank zero as well. -/ |
| theorem exists_uniform_generic_field (r₀ : ℕ) (ε : ℚ) (hε : 0 < ε) : |
| ∃ d : ℕ, 0 < d ∧ r₀ ≤ 3 * d ∧ |
| (∀ r : ℕ, r ≤ r₀ → nongenericFraction (BinaryField d) r ≤ ε) ∧ |
| (∃ S : RankSpace (ZMod 2) (Fin 3 → BinaryField d) r₀, IsGeneric S.1) := by |
| classical |
| have hex (r : ℕ) := exists_generic_subspace_error_numerator (Fin r) |
| choose D hD hbound using hex |
| let M : ℕ := ∑ r ∈ Finset.range (r₀ + 1), D r |
| have hDM (r : ℕ) (hr : r ≤ r₀) : D r ≤ M := by |
| apply Finset.single_le_sum (fun _ _ => Nat.zero_le _) |
| exact Finset.mem_range.mpr (by omega) |
| obtain ⟨N, hN⟩ := exists_nat_gt (max (2 * (M : ℚ)) (2 * (M : ℚ) / ε)) |
| let d := fieldDegree N r₀ |
| have hd : 0 < d := fieldDegree_pos N r₀ |
| have hqN : (N : ℚ) < Fintype.card (BinaryField d) := by |
| exact_mod_cast (by |
| simpa only [Nat.card_eq_fintype_card] using |
| (chosen_binaryField_thresholds N r₀).1) |
| have hq : (0 : ℚ) < Fintype.card (BinaryField d) := by |
| exact_mod_cast Fintype.card_pos |
| have hqM : 2 * (M : ℚ) < Fintype.card (BinaryField d) := |
| (le_max_left _ _).trans_lt (hN.trans hqN) |
| have hqε : 2 * (M : ℚ) / ε < Fintype.card (BinaryField d) := |
| (le_max_right _ _).trans_lt (hN.trans hqN) |
| have hεq : 2 * (M : ℚ) < (Fintype.card (BinaryField d) : ℚ) * ε := |
| (div_lt_iff₀ hε).mp hqε |
| have hsmall (r : ℕ) (hr : r ≤ r₀) : |
| (D r : ℚ) / Fintype.card (BinaryField d) ≤ 1 / 2 ∧ |
| (D r : ℚ) / Fintype.card (BinaryField d) ≤ ε / 2 := by |
| have hDr : (D r : ℚ) ≤ M := by exact_mod_cast hDM r hr |
| constructor |
| · apply (div_le_iff₀ hq).mpr |
| nlinarith |
| · apply (div_le_iff₀ hq).mpr |
| nlinarith |
| have hready (r : ℕ) (hr : r ≤ r₀) := |
| hbound r (BinaryField d) (show (D r : ℚ) / Fintype.card (BinaryField d) < 1 by |
| have := (hsmall r hr).1 |
| linarith) |
| refine ⟨d, hd, rankThreshold_le_triple_fieldDegree N r₀, ?_, ?_⟩ |
| · intro r hr |
| have hb := (hready r hr).2 |
| simp only [Fintype.card_fin] at hb |
| apply hb.trans |
| exact conditional_error_le_of_small _ _ |
| (div_nonneg (Nat.cast_nonneg _) hq.le) (hsmall r hr).1 (hsmall r hr).2 |
| · obtain ⟨S, hS⟩ := (hready r₀ le_rfl).1 |
| refine ⟨⟨S.1, ?_⟩, hS⟩ |
| simpa only [Fintype.card_fin] using S.2 |
|
|
| end |
| |
| end UniqueGamesTheorem.Quadratic |
|
|
| end |
| |
| end OAI |
|
|
