Theorems · Theorem · number theory
le_schnirelmannDensity_iff
∀ {A : Set ℕ} [inst : DecidablePred fun x => x ∈ A] {x : ℝ},
x ≤ schnirelmannDensity A ↔ ∀ (n : ℕ), 0 < n → x ≤ ↑{a ∈ Finset.Ioc 0 n | a ∈ A}.card / ↑n- Defined in
- Mathlib.Combinatorics.Schnirelmann
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DecidablePred
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.rangeproof · cited by 4,705
- Finset.cardstatement and proof · cited by 2,327
- Finset.filterstatement and proof · cited by 949
- Finset.Iocstatement and proof · cited by 301
- Nat.cast_nonneg'proof · cited by 245
- div_nonnegproof · cited by 103
- schnirelmannDensitystatement · cited by 35
- le_ciInf_iffproof · cited by 5
Cited by3
Results whose statement or proof uses this declaration.
- schnirelmannDensity_setOfPred_mod_eq_oneproof · cited by 3
- exists_of_schnirelmannDensity_eq_zeroproof · cited by 0
- schnirelmannDensity_lt_iffproof · cited by 0