| import Mathlib.Algebra.Group.Pointwise.Set.Basic |
| import Mathlib.Data.Nat.Prime.Defs |
| import Mathlib.Data.Set.Finite.Basic |
| import Mathlib.Data.Set.SymmDiff |
| import Mathlib.Order.Interval.Finset.Nat |
|
|
| namespace OAI |
|
|
| namespace Ostmann |
| open scoped Pointwise symmDiff |
|
|
| def primes : Set ℕ := {n | Nat.Prime n} |
|
|
| abbrev sumset (A B : Set ℕ) : Set ℕ := A + B |
|
|
| def EventuallyEqual (S T : Set ℕ) : Prop := |
| ∃ N : ℕ, ∀ n : ℕ, N ≤ n → (n ∈ S ↔ n ∈ T) |
|
|
| def AsymptoticallyIndecomposable : Prop := |
| ∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial → |
| Set.Infinite ((A + B) ∆ {n : ℕ | Nat.Prime n}) |
|
|
| def FiniteModificationIndecomposable : Prop := |
| ∀ S : Set ℕ, (S ∆ primes).Finite → |
| ∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial → A + B ≠ S |
|
|
| @[simp] theorem mem_primes {n : ℕ} : n ∈ primes ↔ Nat.Prime n := Iff.rfl |
|
|
| @[simp] theorem mem_sumset {A B : Set ℕ} {n : ℕ} : |
| n ∈ sumset A B ↔ ∃ a ∈ A, ∃ b ∈ B, a + b = n := Set.mem_add |
|
|
| noncomputable def elementsUpTo (A : Set ℕ) (x : ℕ) : Finset ℕ := by |
| classical |
| exact (Finset.range (x + 1)).filter (fun n => n ∈ A) |
|
|
| noncomputable def countUpTo (A : Set ℕ) (x : ℕ) : ℕ := (elementsUpTo A x).card |
|
|
| @[simp] theorem mem_elementsUpTo {A : Set ℕ} {x n : ℕ} : |
| n ∈ elementsUpTo A x ↔ n ∈ A ∧ n ≤ x := by |
| classical |
| simp only [elementsUpTo, Finset.mem_filter, Finset.mem_range, |
| Nat.lt_succ_iff, and_comm] |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
