File size: 2,504 Bytes
760f90e | 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 | 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 _ _)⟩
/-- Uniform choice of `ell` bounds the trivial character fraction for every
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
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