Loogle!

Result

Found 67 declarations mentioning Finset.Ico, Nat, and Finset.sum.

• Finset.sum_Ico_eq_sum_range 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) (m n : ℕ) : ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range (n - m), f (m + k)
• Finset.sum_range_add_sum_Ico 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {m n : ℕ} (h : m ≤ n) : ∑ k ∈ Finset.range m, f k + ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range n, f k
• Finset.sum_range_eq_add_Ico 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {n : ℕ} (hn : 0 < n) : ∑ x ∈ Finset.range n, f x = f 0 + ∑ x ∈ Finset.Ico 1 n, f x
• Finset.sum_Ico_eq_sub 📋 Mathlib.Algebra.BigOperators.Intervals
  {δ : Type u_3} [AddCommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) : ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range n, f k - ∑ k ∈ Finset.range m, f k
• Finset.sum_Ico_succ_top 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_2} [AddCommMonoid M] {a b : ℕ} (hab : a ≤ b) (f : ℕ → M) : ∑ k ∈ Finset.Ico a (b + 1), f k = ∑ k ∈ Finset.Ico a b, f k + f b
• Finset.sum_eq_sum_Ico_succ_bot 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_2} [AddCommMonoid M] {a b : ℕ} (hab : a < b) (f : ℕ → M) : ∑ k ∈ Finset.Ico a b, f k = f a + ∑ k ∈ Finset.Ico (a + 1) b, f k
• Finset.sum_Ico_sub_bot 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_3} (f : ℕ → M) {m n : ℕ} [AddCommGroup M] (hmn : m < n) : ∑ i ∈ Finset.Ico m n, f i - f m = ∑ i ∈ Finset.Ico (m + 1) n, f i
• Finset.sum_Ico_succ_sub_top 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_3} (f : ℕ → M) {m n : ℕ} [AddCommGroup M] (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m (n + 1), f i - f n = ∑ i ∈ Finset.Ico m n, f i
• Finset.sum_Ico_Ico_comm 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_3} [AddCommMonoid M] (a b : ℕ) (f : ℕ → ℕ → M) : ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico i b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a (j + 1), f i j
• Finset.sum_Ico_Ico_comm' 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_3} [AddCommMonoid M] (a b : ℕ) (f : ℕ → ℕ → M) : ∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico (i + 1) b, f i j = ∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j
• Finset.sum_Ico_consecutive 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_2} [AddCommMonoid M] (f : ℕ → M) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) : ∑ i ∈ Finset.Ico m n, f i + ∑ i ∈ Finset.Ico n k, f i = ∑ i ∈ Finset.Ico m k, f i
• Finset.sum_Ico_sub 📋 Mathlib.Algebra.BigOperators.Intervals
  {M : Type u_3} (f : ℕ → M) {m n : ℕ} [AddCommGroup M] (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m n, (f (i + 1) - f i) = f n - f m
• Finset.sum_Ico_eq_add_neg 📋 Mathlib.Algebra.BigOperators.Intervals
  {δ : Type u_3} [AddCommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) : ∑ k ∈ Finset.Ico m n, f k = ∑ k ∈ Finset.range n, f k + -∑ k ∈ Finset.range m, f k
• Finset.sum_Ico_reflect 📋 Mathlib.Algebra.BigOperators.Intervals
  {δ : Type u_3} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) : ∑ j ∈ Finset.Ico k m, f (n - j) = ∑ j ∈ Finset.Ico (n + 1 - m) (n + 1 - k), f j
• Fin.sum_Ico_cast 📋 Mathlib.Algebra.BigOperators.Fin
  {M : Type u_2} [AddCommMonoid M] {n m : ℕ} (h : n = m) (f : Fin m → M) (a b : Fin n) : ∑ i ∈ Finset.Ico (Fin.cast h a) (Fin.cast h b), f i = ∑ i ∈ Finset.Ico a b, f (Fin.cast h i)
• Fin.sum_Ico_castLE 📋 Mathlib.Algebra.BigOperators.Fin
  {M : Type u_2} [AddCommMonoid M] {n m : ℕ} (h : n ≤ m) (f : Fin m → M) (a b : Fin n) : ∑ i ∈ Finset.Ico (Fin.castLE h a) (Fin.castLE h b), f i = ∑ i ∈ Finset.Ico a b, f (Fin.castLE h i)
• Fin.sum_Ico_castAdd 📋 Mathlib.Algebra.BigOperators.Fin
  {M : Type u_2} [AddCommMonoid M] {n : ℕ} (m : ℕ) (f : Fin (n + m) → M) (a b : Fin n) : ∑ i ∈ Finset.Ico (Fin.castAdd m a) (Fin.castAdd m b), f i = ∑ i ∈ Finset.Ico a b, f (Fin.castAdd m i)
• Fin.sum_Ico_castSucc 📋 Mathlib.Algebra.BigOperators.Fin
  {M : Type u_2} [AddCommMonoid M] {n : ℕ} (f : Fin (n + 1) → M) (a b : Fin n) : ∑ i ∈ Finset.Ico a.castSucc b.castSucc, f i = ∑ i ∈ Finset.Ico a b, f i.castSucc
• Fin.sum_Ico_succ 📋 Mathlib.Algebra.BigOperators.Fin
  {M : Type u_2} [AddCommMonoid M] {n : ℕ} (f : Fin (n + 1) → M) (a b : Fin n) : ∑ i ∈ Finset.Ico a.succ b.succ, f i = ∑ i ∈ Finset.Ico a b, f i.succ
• Nat.geom_sum_Ico_le 📋 Mathlib.Algebra.GeomSum
  {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) : ∑ i ∈ Finset.Ico 1 n, a / b ^ i ≤ a / (b - 1)
• geom_sum_Ico_mul 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} [Ring R] (x : R) {m n : ℕ} (hmn : m ≤ n) : (∑ i ∈ Finset.Ico m n, x ^ i) * (x - 1) = x ^ n - x ^ m
• geom_sum_Ico_mul_neg 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} [Ring R] (x : R) {m n : ℕ} (hmn : m ≤ n) : (∑ i ∈ Finset.Ico m n, x ^ i) * (1 - x) = x ^ m - x ^ n
• geom_sum_Ico 📋 Mathlib.Algebra.GeomSum
  {K : Type u_2} [DivisionRing K] {x : K} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1)
• geom_sum_Ico' 📋 Mathlib.Algebra.GeomSum
  {K : Type u_2} [DivisionRing K] {x : K} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x)
• geom_sum_Ico_le_of_lt_one 📋 Mathlib.Algebra.GeomSum
  {K : Type u_2} [Field K] [LinearOrder K] [IsStrictOrderedRing K] {x : K} (hx : 0 ≤ x) (h'x : x < 1) {m n : ℕ} : ∑ i ∈ Finset.Ico m n, x ^ i ≤ x ^ m / (1 - x)
• Commute.geom_sum₂_Ico_mul 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} [Ring R] {x y : R} (h : Commute x y) {m n : ℕ} (hmn : m ≤ n) : (∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ (n - m) * x ^ m
• Commute.mul_geom_sum₂_Ico 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} [Ring R] {x y : R} (h : Commute x y) {m n : ℕ} (hmn : m ≤ n) : (x - y) * ∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i) = x ^ n - x ^ m * y ^ (n - m)
• mul_geom_sum₂_Ico 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} [CommRing R] (x y : R) {m n : ℕ} (hmn : m ≤ n) : (x - y) * ∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i) = x ^ n - x ^ m * y ^ (n - m)
• Commute.geom_sum₂_Ico 📋 Mathlib.Algebra.GeomSum
  {K : Type u_2} [DivisionRing K] {x y : K} (h : Commute x y) (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m) / (x - y)
• geom_sum₂_Ico 📋 Mathlib.Algebra.GeomSum
  {K : Type u_2} [Field K] {x y : K} (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) : ∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m) / (x - y)
• Nat.Prime.emultiplicity_factorial 📋 Mathlib.Data.Nat.Multiplicity
  {p : ℕ} (hp : Nat.Prime p) {n b : ℕ} : Nat.log p n < b → emultiplicity p n.factorial = ↑(∑ i ∈ Finset.Ico 1 b, n / p ^ i)
• Nat.Prime.pow_dvd_factorial_iff 📋 Mathlib.Data.Nat.Multiplicity
  {p n r b : ℕ} (hp : Nat.Prime p) (hbn : Nat.log p n < b) : p ^ r ∣ n.factorial ↔ r ≤ ∑ i ∈ Finset.Ico 1 b, n / p ^ i
• Nat.Prime.multiplicity_choose_aux 📋 Mathlib.Data.Nat.Multiplicity
  {p n b k : ℕ} (hp : Nat.Prime p) (hkn : k ≤ n) : ∑ i ∈ Finset.Ico 1 b, n / p ^ i = ∑ i ∈ Finset.Ico 1 b, k / p ^ i + ∑ i ∈ Finset.Ico 1 b, (n - k) / p ^ i + {i ∈ Finset.Ico 1 b | p ^ i ≤ k % p ^ i + (n - k) % p ^ i}.card
• Finset.sum_Ico_by_parts 📋 Mathlib.Algebra.BigOperators.Module
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ} (hmn : m < n) : ∑ i ∈ Finset.Ico m n, f i • g i = f (n - 1) • ∑ i ∈ Finset.range n, g i - f m • ∑ i ∈ Finset.range m, g i - ∑ i ∈ Finset.Ico m (n - 1), (f (i + 1) - f i) • ∑ i ∈ Finset.range (i + 1), g i
• edist_le_Ico_sum_edist 📋 Mathlib.Topology.EMetricSpace.Basic
  {α : Type u} [PseudoEMetricSpace α] (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1))
• edist_le_Ico_sum_of_edist_le 📋 Mathlib.Topology.EMetricSpace.Basic
  {α : Type u} [PseudoEMetricSpace α] {f : ℕ → α} {m n : ℕ} (hmn : m ≤ n) {d : ℕ → ENNReal} (hd : ∀ {k : ℕ}, m ≤ k → k < n → edist (f k) (f (k + 1)) ≤ d k) : edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i
• dist_le_Ico_sum_dist 📋 Mathlib.Topology.MetricSpace.Pseudo.Basic
  {α : Type u} [PseudoMetricSpace α] (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))
• dist_le_Ico_sum_of_dist_le 📋 Mathlib.Topology.MetricSpace.Pseudo.Basic
  {α : Type u} [PseudoMetricSpace α] {f : ℕ → α} {m n : ℕ} (hmn : m ≤ n) {d : ℕ → ℝ} (hd : ∀ {k : ℕ}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i
• FormalMultilinearSeries.comp_partialSum 📋 Mathlib.Analysis.Analytic.Composition
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (M N : ℕ) (z : E) : q.partialSum M (∑ i ∈ Finset.Ico 1 N, (p i) fun _j => z) = ∑ i ∈ FormalMultilinearSeries.compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j => z
• FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1 📋 Mathlib.Analysis.Analytic.Inverse
  (n : ℕ) (p : ℕ → ℝ) (hp : ∀ (k : ℕ), 0 ≤ p k) {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) : ∑ k ∈ Finset.Ico 2 (n + 1), a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j, p (c.blocksFun j) ≤ ∑ j ∈ Finset.Ico 2 (n + 1), r ^ j * (∑ k ∈ Finset.Ico 1 n, a ^ k * p k) ^ j
• FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos_aux2 📋 Mathlib.Analysis.Analytic.Inverse
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ} (hn : 2 ≤ n + 1) (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) {r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) (hC : 0 ≤ C) (hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n) : ∑ k ∈ Finset.Ico 1 (n + 1), a ^ k * ‖p.rightInv i x k‖ ≤ ‖↑i.symm‖ * a + ‖↑i.symm‖ * C * ∑ k ∈ Finset.Ico 2 (n + 1), (r * ∑ j ∈ Finset.Ico 1 n, a ^ j * ‖p.rightInv i x j‖) ^ k
• eVariationOn.sum_le_of_monotoneOn_Icc 📋 Mathlib.Topology.EMetricSpace.BoundedVariation
  {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α} (hu : MonotoneOn u (Set.Icc m n)) (us : ∀ i ∈ Set.Icc m n, u i ∈ s) : ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s
• intervalIntegral.sum_integral_adjacent_intervals_Ico 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
  {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n) (hint : ∀ k ∈ Set.Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))) : ∑ k ∈ Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a n, f x ∂μ
• Finset.le_sum_schlomilch' 📋 Mathlib.Analysis.PSeries
  {M : Type u_1} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] {f : ℕ → M} {u : ℕ → ℕ} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu : Monotone u) (n : ℕ) : ∑ k ∈ Finset.Ico (u 0) (u n), f k ≤ ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u k)
• Finset.le_sum_condensed' 📋 Mathlib.Analysis.PSeries
  {M : Type u_1} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) : ∑ k ∈ Finset.Ico 1 (2 ^ n), f k ≤ ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ k)
• Finset.sum_condensed_le' 📋 Mathlib.Analysis.PSeries
  {M : Type u_1} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) : ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ (k + 1)) ≤ ∑ k ∈ Finset.Ico 2 (2 ^ n + 1), f k
• Finset.sum_schlomilch_le' 📋 Mathlib.Analysis.PSeries
  {M : Type u_1} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] {f : ℕ → M} {u : ℕ → ℕ} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu : Monotone u) (n : ℕ) : ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Finset.Ico (u 0 + 1) (u n + 1), f k
• Finset.sum_condensed_le 📋 Mathlib.Analysis.PSeries
  {M : Type u_1} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) : ∑ k ∈ Finset.range (n + 1), 2 ^ k • f (2 ^ k) ≤ f 1 + 2 • ∑ k ∈ Finset.Ico 2 (2 ^ n + 1), f k
• Finset.sum_schlomilch_le 📋 Mathlib.Analysis.PSeries
  {M : Type u_1} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] {f : ℕ → M} {u : ℕ → ℕ} {C : ℕ} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (h_nonneg : ∀ (n : ℕ), 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) (n : ℕ) : ∑ k ∈ Finset.range (n + 1), (u (k + 1) - u k) • f (u k) ≤ (u 1 - u 0) • f (u 0) + C • ∑ k ∈ Finset.Ico (u 0 + 1) (u n + 1), f k
• padicValNat_factorial 📋 Mathlib.NumberTheory.Padics.PadicVal.Basic
  {p n b : ℕ} [hp : Fact (Nat.Prime p)] (hnb : Nat.log p n < b) : padicValNat p n.factorial = ∑ i ∈ Finset.Ico 1 b, n / p ^ i
• AntitoneOn.integral_le_sum_Ico 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) : ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ x ∈ Finset.Ico a b, f ↑x
• MonotoneOn.sum_le_integral_Ico 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) : ∑ x ∈ Finset.Ico a b, f ↑x ≤ ∫ (x : ℝ) in ↑a..↑b, f x
• AntitoneOn.sum_le_integral_Ico 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) : ∑ i ∈ Finset.Ico a b, f ↑(i + 1) ≤ ∫ (x : ℝ) in ↑a..↑b, f x
• MonotoneOn.integral_le_sum_Ico 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) : ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ i ∈ Finset.Ico a b, f ↑(i + 1)
• integral_le_sum_Ico_of_le 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (h : ∀ i ∈ Set.Ico a b, ∀ x ∈ Set.Ico ↑i ↑(i + 1), g x ≤ f ↑i) (hg : MeasureTheory.IntegrableOn g (Set.Ico ↑a ↑b) MeasureTheory.volume) : ∫ (x : ℝ) in ↑a..↑b, g x ≤ ∑ i ∈ Finset.Ico a b, f ↑i
• sum_Ico_le_integral_of_le 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (h : ∀ i ∈ Set.Ico a b, ∀ x ∈ Set.Ico ↑i ↑(i + 1), f ↑i ≤ g x) (hg : MeasureTheory.IntegrableOn g (Set.Ico ↑a ↑b) MeasureTheory.volume) : ∑ i ∈ Finset.Ico a b, f ↑i ≤ ∫ (x : ℝ) in ↑a..↑b, g x
• integral_le_sum_mul_Ico_of_antitone_monotone 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) (hg : MonotoneOn g (Set.Icc (↑a - 1) (↑b - 1))) (fpos : 0 ≤ f ↑b) (gpos : 0 ≤ g (↑a - 1)) : ∫ (x : ℝ) in ↑a..↑b, f x * g (x - 1) ≤ ∑ i ∈ Finset.Ico a b, f ↑i * g ↑i
• sum_mul_Ico_le_integral_of_monotone_antitone 📋 Mathlib.Analysis.SumIntegralComparisons
  {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) (hg : AntitoneOn g (Set.Icc (↑a - 1) (↑b - 1))) (fpos : 0 ≤ f ↑a) (gpos : 0 ≤ g (↑b - 1)) : ∑ i ∈ Finset.Ico a b, f ↑i * g ↑i ≤ ∫ (x : ℝ) in ↑a..↑b, f x * g (x - 1)
• AkraBazziRecurrence.sumTransform_def 📋 Mathlib.Computability.AkraBazzi.AkraBazzi
  {p : ℝ} {g : ℝ → ℝ} {n₀ n : ℕ} : AkraBazziRecurrence.sumTransform p g n₀ n = ↑n ^ p * ∑ u ∈ Finset.Ico n₀ n, g ↑u / ↑u ^ (p + 1)
• AkraBazziRecurrence.sumTransform.eq_1 📋 Mathlib.Computability.AkraBazzi.AkraBazzi
  (p : ℝ) (g : ℝ → ℝ) (n₀ n : ℕ) : AkraBazziRecurrence.sumTransform p g n₀ n = ↑n ^ p * ∑ u ∈ Finset.Ico n₀ n, g ↑u / ↑u ^ (p + 1)
• sum_Ico_pow 📋 Mathlib.NumberTheory.Bernoulli
  (n p : ℕ) : ∑ k ∈ Finset.Ico 1 (n + 1), ↑k ^ p = ∑ i ∈ Finset.range (p + 1), bernoulli' i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
• ZMod.eisenstein_lemma 📋 Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas
  {p : ℕ} [Fact (Nat.Prime p)] (hp : p ≠ 2) {a : ℕ} (ha1 : a % 2 = 1) (ha0 : ↑a ≠ 0) : legendreSym p ↑a = (-1) ^ ∑ x ∈ Finset.Ico 1 (p / 2).succ, x * a / p
• ZMod.sum_mul_div_add_sum_mul_div_eq_mul 📋 Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas
  (p q : ℕ) [hp : Fact (Nat.Prime p)] (hq0 : ↑q ≠ 0) : ∑ a ∈ Finset.Ico 1 (p / 2).succ, a * q / p + ∑ a ∈ Finset.Ico 1 (q / 2).succ, a * p / q = p / 2 * (q / 2)
• ZMod.eisenstein_lemma_aux 📋 Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas
  (p : ℕ) [Fact (Nat.Prime p)] [Fact (p % 2 = 1)] {a : ℕ} (ha2 : a % 2 = 1) (hap : ↑a ≠ 0) : {x ∈ Finset.Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}.card ≡ ∑ x ∈ Finset.Ico 1 (p / 2).succ, x * a / p [MOD 2]
• MeasureTheory.stoppedValue_sub_eq_sum 📋 Mathlib.Probability.Process.Stopping
  {Ω : Type u_1} {β : Type u_2} {u : ℕ → Ω → β} {τ π : Ω → ℕ} [AddCommGroup β] (hle : τ ≤ π) : MeasureTheory.stoppedValue u π - MeasureTheory.stoppedValue u τ = fun ω => (∑ i ∈ Finset.Ico (τ ω) (π ω), (u (i + 1) - u i)) ω
• MeasureTheory.upcrossingsBefore_eq_sum 📋 Mathlib.Probability.Martingale.Upcrossing
  {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hab : a < b) : MeasureTheory.upcrossingsBefore a b f N ω = ∑ i ∈ Finset.Ico 1 (N + 1), {n | MeasureTheory.upperCrossingTime a b f N n ω < N}.indicator 1 i
• PowerSeries.trunc.eq_1 📋 Mathlib.RingTheory.PowerSeries.Trunc
  {R : Type u_1} [Semiring R] (n : ℕ) (φ : PowerSeries R) : PowerSeries.trunc n φ = ∑ m ∈ Finset.Ico 0 n, (Polynomial.monomial m) ((PowerSeries.coeff R m) φ)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision b5368c7