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92416a2 | 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 87 88 | 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
/-!
# One field for every rank up to a fixed threshold
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 _
/-- Simultaneous quantitative genericity, with the precise threshold order
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
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