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2b3d0c0 | 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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Order.Interval.Finset.Nat
import OAI.NumberTheory.Ostmann.Defs
namespace OAI
/-!
# Sets in Ostmann's inverse Goldbach problem
We use `ℕ` for the manuscript's nonnegative integers. The threshold convention
uses `N ≤ n`; changing finitely many initial values does not affect it.
-/
namespace Ostmann
/-- Both prime coverage and exclusion of composite sums beyond one threshold. -/
def EventuallyPrimeSumset (A B : Set ℕ) : Prop :=
∃ N : ℕ, ∀ n, N ≤ n → (n ∈ sumset A B ↔ n.Prime)
theorem sumset_comm (A B : Set ℕ) : sumset A B = sumset B A := by
ext n
simp only [mem_sumset]
constructor <;> rintro ⟨a, ha, b, hb, hab⟩
· exact ⟨b, hb, a, ha, (Nat.add_comm b a).trans hab⟩
· exact ⟨b, hb, a, ha, (Nat.add_comm b a).trans hab⟩
theorem EventuallyPrimeSumset.symm {A B : Set ℕ}
(h : EventuallyPrimeSumset A B) : EventuallyPrimeSumset B A := by
simpa only [EventuallyPrimeSumset, sumset_comm B A] using h
open scoped symmDiff
/-- The manuscript's finite symmetric difference and threshold formulations agree. -/
theorem finite_symmDiff_iff (A B : Set ℕ) :
(sumset A B ∆ primes).Finite ↔ EventuallyPrimeSumset A B := by
constructor
· intro h
obtain ⟨M, hM⟩ := Set.finite_iff_bddAbove.mp h
refine ⟨M + 1, ?_⟩
intro n hn
have hout : n ∉ sumset A B ∆ primes := by
intro hin
have := hM hin
omega
simp only [Set.mem_symmDiff, primes, Set.mem_ofPred_eq] at hout
tauto
· rintro ⟨N, hN⟩
apply (Set.finite_Iio N).subset
intro n hn
by_contra hlt
have heq := hN n (Nat.le_of_not_gt hlt)
simp only [Set.mem_symmDiff, primes, Set.mem_ofPred_eq, heq,
and_not_self, or_self] at hn
/-- Each sufficiently large sum of elements of the summands is prime. -/
theorem EventuallyPrimeSumset.prime_add {A B : Set ℕ}
(h : EventuallyPrimeSumset A B) :
∃ N, ∀ a ∈ A, ∀ b ∈ B, N ≤ a + b → (a + b).Prime := by
obtain ⟨N, hN⟩ := h
exact ⟨N, fun a ha b hb hab => (hN (a + b) hab).mp ⟨a, ha, b, hb, rfl⟩⟩
/-- The two infinite summands formulation, stated independently of its proof. -/
def TwoInfiniteSummandsImpossible : Prop :=
∀ A B : Set ℕ, A.Infinite → B.Infinite → ¬ EventuallyPrimeSumset A B
/-- Ostmann's main statement, including all finite modifications of the primes. -/
def InverseGoldbach : Prop :=
∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial → (sumset A B ∆ primes).Infinite
end Ostmann
end OAI
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