| import OAI.Computability.UniqueGames.Quadratic.AlignmentCertificateLemmas |
| import OAI.Computability.UniqueGames.Quadratic.Block |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| # The binary trace annihilator of a quadratic-block hyperplane |
|
|
| The trace annihilator of the field hyperplane perpendicular to a nonzero |
| vector is exactly its field line. Scaling test vectors first upgrades binary |
| trace orthogonality to field-valued orthogonality; explicit coordinate test |
| vectors then identify the annihilator. No earlier gadget result is used. |
| -/ |
|
|
| namespace UniqueGamesTheorem.Quadratic |
|
|
| variable {F : Type*} [Field F] |
|
|
| private theorem dot_sub_right (v y z : Vec F) : |
| dot v (y - z) = dot v y - dot v z := by |
| simp [dot] |
| ring |
|
|
| private theorem dot_single (v : Vec F) (i : Fin 3) (a : F) : |
| dot v (Pi.single i a) = v i * a := by |
| fin_cases i <;> simp [dot] |
|
|
| / |
| hyperplane is the line generated by that vector. -/ |
| theorem annihilator_hyperplane_iff_mem_line {v z : Vec F} (hv : v ≠ 0) : |
| (∀ y ∈ hyperplane v, dot z y = 0) ↔ z ∈ line v := by |
| constructor |
| · intro hz |
| have hv' : ∃ i : Fin 3, v i ≠ 0 := by |
| by_contra! h |
| exact hv (funext h) |
| obtain ⟨i, hi⟩ := hv' |
| apply (mem_line_iff v z).mpr |
| refine ⟨z i / v i, ?_⟩ |
| ext j |
| have hy : Pi.single j (1 : F) - (v j / v i) • Pi.single i (1 : F) ∈ |
| hyperplane v := by |
| rw [mem_hyperplane, dot_sub_right, dot_smul_right, dot_single, dot_single] |
| simp [div_mul_cancel₀ _ hi] |
| have hzero := hz _ hy |
| rw [dot_sub_right, dot_smul_right, dot_single, dot_single, |
| mul_one, mul_one, sub_eq_zero] at hzero |
| change (z i / v i) * v j = z j |
| rw [hzero] |
| simp only [div_eq_mul_inv] |
| ring |
| · intro hz y hy |
| obtain ⟨t, rfl⟩ := (mem_line_iff v z).mp hz |
| rw [dot_smul_left, (mem_hyperplane v y).mp hy, mul_zero] |
|
|
| variable [Finite F] [Algebra (ZMod 2) F] |
|
|
| / |
| nondegeneracy is provided by the proved finite-field trace pairing theorem. -/ |
| theorem trace_annihilator_hyperplane_iff_mem_line {v z : Vec F} (hv : v ≠ 0) : |
| (∀ y ∈ hyperplane v, traceBinary (dot z y) = 0) ↔ z ∈ line v := by |
| rw [← annihilator_hyperplane_iff_mem_line hv] |
| constructor |
| · intro hz y hy |
| apply (trace_mul_vanish_iff (dot z y)).mp |
| intro t |
| have ht := hz (t • y) ((hyperplane v).smul_mem t hy) |
| simpa only [dot_smul_right] using ht |
| · intro hz y hy |
| rw [hz y hy, map_zero] |
|
|
| / |
| theorem trace_annihilator_hyperplane (v : Vec F) (hv : v ≠ 0) : |
| {z : Vec F | ∀ y ∈ hyperplane v, Algebra.trace (ZMod 2) F (dot z y) = 0} = |
| (line v : Set (Vec F)) := by |
| ext z |
| exact trace_annihilator_hyperplane_iff_mem_line hv |
|
|
| end UniqueGamesTheorem.Quadratic |
|
|
| end |
|
|
| end OAI |
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|
