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0789dda | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | 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
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