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ea76aac | 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 | import Mathlib
import OAI.Computability.MaxCut.Games.MatrixGap
import OAI.Computability.MaxCut.Games.AffineWitness
namespace OAI
namespace MaxCutGames.Inverse.AffineWitness
theorem first_bit_witness {m ℓ r : ℕ}
(e : (Fin m → F2) →ₗ[F2] F2)
(A : (Fin ℓ → F2) →ₗ[F2] (Fin r → F2))
(Z : Submodule F2 (Fin m → F2))
(M₀ : (Fin m → F2) →ₗ[F2] (Fin ℓ → F2))
(z : Fin m → F2) (u : Fin ℓ → F2)
(F : ((Fin m → F2) →ₗ[F2] (Fin ℓ → F2)) → (Fin ℓ → F2))
(hfold : ∀ (h : A.ker) M, F (shift e h M) = F M + h)
(α : ℝ) (hα : 0 < α)
(hgood : α / 2 ≤
(Nat.card {M : Slice A Z M₀ // F M = M.val z + u} : ℝ) /
(Nat.card (Slice A Z M₀) : ℝ))
(hsmall : 1 / (2 : ℝ) ^ (ℓ - r) < α / 8) :
∃ z' : Fin m → F2, ∃ u' : Fin ℓ → F2, e z' = 1 ∧
∀ M, InSlice A Z M₀ M → M z' + u' = M z + u := by
have hpow : (0 : ℝ) < 2 ^ (ℓ - r) := pow_pos (by norm_num) _
have hsep := (div_lt_iff₀ hpow).mp hsmall
have hk : (2 : ℝ) ^ (ℓ - r) ≤ (Nat.card A.ker : ℝ) := by
exact_mod_cast binary_kernel_card_lower A
have hscale := mul_le_mul_of_nonneg_left hk hα.le
have hsize : 1 < (α / 2) * (Nat.card A.ker : ℝ) := by
nlinarith only [hsep, hscale]
exact normalize_of_density e A Z M₀ z u F hfold (α / 2) hgood hsize
end MaxCutGames.Inverse.AffineWitness
end OAI
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