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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 | import Mathlib
namespace OAI
open scoped Pointwise symmDiff
namespace Ostmann
def primes : Set ℕ := {n | Nat.Prime n}
abbrev sumset (A B : Set ℕ) : Set ℕ := A + B
def EventuallyPrimeSumset (A B : Set ℕ) : Prop :=
∃ N : ℕ, ∀ n, N ≤ n → (n ∈ sumset A B ↔ n.Prime)
def TwoInfiniteSummandsImpossible : Prop :=
∀ A B : Set ℕ, A.Infinite → B.Infinite → ¬ EventuallyPrimeSumset A B
def InverseGoldbach : Prop :=
∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial →
Set.Infinite (@symmDiff (Set ℕ)
(@SemilatticeSup.toMax (Set ℕ)
(@Lattice.toSemilatticeSup (Set ℕ)
(@CompleteLattice.toLattice (Set ℕ)
(@CompleteBooleanAlgebra.toCompleteLattice (Set ℕ)
(@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra (Set ℕ)
(@Set.instCompleteAtomicBooleanAlgebra ℕ))))))
Set.instSDiff (sumset A B) primes)
theorem inverseGoldbach : InverseGoldbach := by
sorry
theorem twoInfiniteSummandsImpossible : TwoInfiniteSummandsImpossible := by
sorry
end Ostmann
end OAI
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