Theorems · Theorem · order theory
Finset.sum_nonneg
∀ {ι : Type u_1} {N : Type u_5} [inst : AddCommMonoid N] [inst_1 : Preorder N] {f : ι → N} {s : Finset ι}
[AddLeftMono N], (∀ i ∈ s, 0 ≤ f i) → 0 ≤ ∑ i ∈ s, f i- Cited by
- 91 results in Mathlib
- Foundations
- Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- AddCommMonoidstatement and proof · cited by 12,281
- Preorderstatement and proof · cited by 7,952
- Finset.sumstatement · cited by 5,195
- le_reflproof · cited by 2,061
- le_transproof · cited by 985
- AddLeftMonostatement and proof · cited by 687
- Finset.sum_const_zeroproof · cited by 219
- Finset.sum_le_sumproof · cited by 182
Cited by91
Results whose statement or proof uses this declaration.
- Finset.sum_le_sum_of_subset_of_nonnegproof · cited by 31
- Finset.sum_eq_zero_iff_of_nonnegproof · cited by 19
- Function.hasTemperateGrowth_one_add_norm_sq_rpowproof · cited by 9
- summable_norm_iffproof · cited by 8
- tsum_of_norm_boundedproof · cited by 7
- Finsupp.sum_nonnegproof · cited by 6
- Real.geom_mean_le_arith_mean_weightedproof · cited by 5
- ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_of_le_oneproof · cited by 5
- Fintype.sum_nonnegproof · cited by 4
- Convex.centerMass_memproof · cited by 4
- Finset.sum_sq_le_sum_mul_sum_of_sq_le_mulproof · cited by 3
- Matrix.posSemidef_conjTranspose_mul_selfproof · cited by 3