Theorems · Theorem · number theory
schnirelmannDensity_eq_one_iff
∀ {A : Set ℕ} [inst : DecidablePred fun x => x ∈ A], schnirelmannDensity A = 1 ↔ {0}ᶜ ⊆ AThe Schnirelmann density of A is 1 if and only if A contains all the positive naturals.
- Defined in
- Mathlib.Combinatorics.Schnirelmann
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 121 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.
Cites23
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 · cited by 25,697
- Finsetproof · cited by 13,712
- Compl.complstatement and proof · cited by 2,925
- Finset.cardproof · cited by 2,327
- le_reflproof · cited by 2,061
- LT.lt.ne'proof · cited by 1,417
- Finset.filterproof · cited by 949
- LE.le.trans_ltproof · cited by 795
- not_leproof · cited by 328
- Finset.Iocproof · cited by 301
- Nat.cast_leproof · cited by 159
Cited by1
Results whose statement or proof uses this declaration.
- schnirelmannDensity_eq_one_iff_of_zero_memproof · cited by 1