| 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 |
|
|
