Theorems · Theorem · real analysis
AntitoneOn.integral_le_sum
∀ {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ},
AntitoneOn f (Set.Icc x₀ (x₀ + ↑a)) → ∫ (x : ℝ) in x₀..x₀ + ↑a, f x ≤ ∑ i ∈ Finset.range a, f (x₀ + ↑i)- Defined in
- Mathlib.Analysis.SumIntegralComparisons
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupproof · cited by 15,752
- NormedSpaceproof · cited by 12,499
- MeasureTheory.Measureproof · cited by 10,939
- Finset.sumstatement and proof · cited by 5,195
- one_mulproof · cited by 2,841
- add_zeroproof · cited by 2,707
- Nat.cast_oneproof · cited by 2,501
- Finset.sum_congrproof · cited by 2,323
- Set.Iccstatement and proof · cited by 1,702
- Finset.rangestatement and proof · cited by 1,341
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
Cited by3
Results whose statement or proof uses this declaration.
- AntitoneOn.integral_le_sum_Icoproof · cited by 3
- AntitoneOn.integrableOn_Ioi_of_summable_comp_addproof · cited by 2
- MonotoneOn.sum_le_integralproof · cited by 0