| import Mathlib | |
| namespace OAI | |
| namespace GaussianMoat | |
| abbrev PrimeVertex := {z : GaussianInt // Irreducible z} | |
| noncomputable def primeGraph (D : ℝ) : SimpleGraph PrimeVertex where | |
| Adj p q := p ≠ q ∧ dist (p.val : ℂ) (q.val : ℂ) ≤ D | |
| symm := by | |
| constructor | |
| intro p q h | |
| exact ⟨h.1.symm, by simpa only [dist_comm] using h.2⟩ | |
| loopless := by | |
| constructor | |
| intro p h | |
| exact h.1 rfl | |
| def component (D : ℝ) (p : PrimeVertex) : Set PrimeVertex := | |
| {q | (primeGraph D).Reachable p q} | |
| def MainEndpoint : Prop := | |
| ∀ (D : ℝ) (z : ℕ → GaussianInt), | |
| (∀ t, Irreducible (z t)) → Function.Injective z → | |
| (∀ t, dist (z (t + 1) : ℂ) (z t : ℂ) ≤ D) → False | |
| def UniformEndpoint : Prop := | |
| ∀ D : ℝ, ∃ B : ℕ, | |
| (∀ p : PrimeVertex, (component D p).Finite ∧ (component D p).ncard ≤ B) ∧ | |
| (∀ (n : ℕ) (z : Fin n → GaussianInt), | |
| (∀ i, Irreducible (z i)) → Function.Injective z → | |
| (∀ i j : Fin n, j.val = i.val + 1 → dist (z j : ℂ) (z i : ℂ) ≤ D) → | |
| n ≤ B) | |
| theorem fullMain : MainEndpoint ∧ UniformEndpoint := by | |
| sorry | |
| end GaussianMoat | |
| end OAI | |