| import OAI.Geometry.HeilbronnTriangle.AuxiliarySet | |
| namespace OAI | |
| namespace Problem355.AuxiliarySet | |
| open scoped BigOperators | |
| theorem exists_outcome_of_inclusion_sum_pos | |
| {Θ : Type*} (T : Finset Θ) (μ : Θ → ℝ) (P : Θ → Prop) | |
| [DecidablePred P] (hpositive : 0 < ∑ θ ∈ T, if P θ then μ θ else 0) : | |
| ∃ θ ∈ T, P θ := by | |
| classical | |
| by_contra hnone | |
| push Not at hnone | |
| have hzero : (∑ θ ∈ T, if P θ then μ θ else 0) = 0 := by | |
| apply Finset.sum_eq_zero | |
| intro θ hθ | |
| simp [hnone θ hθ] | |
| rw [hzero] at hpositive | |
| exact lt_irrefl 0 hpositive | |
| theorem matrix_support_of_inclusion_sum_pos | |
| {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) (ZMod q)) | |
| [DecidablePred (fun θ => ∀ j, (fun i => A i j) ∈ transformed S (a θ) (G θ))] | |
| (hpositive : 0 < ∑ θ ∈ T, | |
| if ∀ j, (fun i => A i j) ∈ transformed S (a θ) (G θ) then μ θ else 0) : | |
| (∀ j, (fun i => A i j) ≠ 0) ∧ | |
| (AffineIndependent (ZMod q) (fun j => fun i => A i j) ∨ | |
| ∃ j k, j ≠ k ∧ (fun i => A i j) = (fun i => A i k)) ∧ | |
| (AffineIndependent (ZMod q) (fun j => fun i => A i j) → | |
| ∀ x : Fin 3 → ℤ, (∀ i, |x i| ≤ (H : ℤ)) → x ≠ 0 → | |
| A.mulVec (fun i => (x i : ZMod q)) ≠ 0) := by | |
| classical | |
| obtain ⟨θ, hθ, hcols⟩ := exists_outcome_of_inclusion_sum_pos T μ _ hpositive | |
| have hcap := isCap_transformed hS (a θ) (G θ) | |
| have hzero := zero_not_mem_of_allowed hH hbox (a θ) (ha θ hθ) (G θ) | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro j hj | |
| exact hzero (hj ▸ hcols j) | |
| · by_cases hinj : Function.Injective (fun j => fun i => A i j) | |
| · exact Or.inl (affineIndependent_of_mem_cap hcap hcols hinj) | |
| · right | |
| simp only [Function.Injective] at hinj | |
| push Not at hinj | |
| obtain ⟨j, k, heq, hne⟩ := hinj | |
| exact ⟨j, k, hne, heq⟩ | |
| · intro hAI x hx hne hrel | |
| have hsum := short_relation_excluded hS hbox (a θ) (ha θ hθ) (G θ) | |
| hcols hAI.injective hx hHq hne | |
| apply hsum | |
| funext i | |
| have hi := congrFun hrel i | |
| simpa [Matrix.mulVec, dotProduct, Finset.sum_apply, mul_comm] using hi | |
| end Problem355.AuxiliarySet | |
| end OAI | |