| import OAI.Geometry.HeilbronnTriangle.AuxiliarySupport |
| import OAI.Geometry.HeilbronnTriangle.PrimitiveNormal |
|
|
| namespace OAI |
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|
| namespace Problem355.AuxiliarySet |
|
|
| open scoped BigOperators |
|
|
| theorem coordinate_abs_le_of_euclidean_norm_le |
| {x : PrimitiveNormal.Vector} {H : ℕ} |
| (hx : ‖PrimitiveNormal.toEuclidean x‖ ≤ (H : ℝ)) : |
| ∀ i, |x i| ≤ (H : ℤ) := by |
| intro i |
| have hi := (PiLp.norm_apply_le (PrimitiveNormal.toEuclidean x) i).trans hx |
| have hi' : |(x i : ℝ)| ≤ (H : ℝ) := by |
| simpa [PrimitiveNormal.toEuclidean, Real.norm_eq_abs] using hi |
| exact_mod_cast hi' |
|
|
| theorem cast_mulVec_eq_zero {q : ℕ} |
| {A : Matrix (Fin 3) (Fin 3) ℤ} {x : Fin 3 → ℤ} |
| (hx : A.mulVec x = 0) : |
| (A.map (Int.castRingHom (ZMod q))).mulVec (fun i => (x i : ZMod q)) = 0 := by |
| funext i |
| have hi := congrArg (fun z : ℤ => (z : ZMod q)) (congrFun hx i) |
| simpa [Matrix.mulVec, dotProduct, Matrix.map_apply] using hi |
|
|
| theorem kernel_norm_gt_of_positive_auxiliary_inclusion |
| {q H w : ℕ} [Fact q.Prime] (hH : 1 ≤ H) (hHq : H < q) |
| {S : Finset (Fin 3 → ZMod q)} |
| (hS : AuxiliaryCap.IsCap (S : Set (Fin 3 → ZMod q))) |
| (hbox : ∀ v ∈ S, (v 0).val < w) |
| {Θ : Type*} (T : Finset Θ) (μ : Θ → ℝ) |
| (a : Θ → Fin 3 → ZMod q) |
| (G : Θ → ((Fin 3 → ZMod q) ≃ₗ[ZMod q] (Fin 3 → ZMod q))) |
| (ha : ∀ θ ∈ T, ∀ n : ℕ, 1 ≤ n → n ≤ 3 * H → |
| (n : ZMod q) * a θ 0 ∉ Auxiliary.integerWindow q (3 * H * w)) |
| (A : Matrix (Fin 3) (Fin 3) ℤ) |
| [DecidablePred (fun θ => |
| ∀ j, (fun i => (A i j : ZMod q)) ∈ transformed S (a θ) (G θ))] |
| (hpositive : 0 < ∑ θ ∈ T, |
| if ∀ j, (fun i => (A i j : ZMod q)) ∈ transformed S (a θ) (G θ) |
| then μ θ else 0) |
| (hAI : AffineIndependent (ZMod q) (fun j => fun i => (A i j : ZMod q))) |
| {x : PrimitiveNormal.Vector} (hne : x ≠ 0) (hkernel : A.mulVec x = 0) : |
| (H : ℝ) < ‖PrimitiveNormal.toEuclidean x‖ := by |
| classical |
| by_contra hnot |
| have hnorm : ‖PrimitiveNormal.toEuclidean x‖ ≤ (H : ℝ) := le_of_not_gt hnot |
| have hsupport := matrix_support_of_inclusion_sum_pos hH hHq hS hbox T μ a G ha |
| (fun i j => (A i j : ZMod q)) hpositive |
| have hno := hsupport.2.2 hAI x (coordinate_abs_le_of_euclidean_norm_le hnorm) hne |
| exact hno (cast_mulVec_eq_zero hkernel) |
|
|
| end Problem355.AuxiliarySet |
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| end OAI |
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