Loogle!

Result

Found 9 declarations mentioning Even and Finset.sum.

• Finset.even_sum 📋 Mathlib.Algebra.BigOperators.Group.Finset.Lemmas
  {ι : Type u_1} {α : Type u_2} {s : Finset ι} [AddCommMonoid α] (f : ι → α) (h : ∀ c ∈ s, Even (f c)) : Even (∑ i ∈ s, f i)
• Fin.sum_neg_one_pow 📋 Mathlib.Algebra.BigOperators.Fin
  (R : Type u_3) [Ring R] (m : ℕ) : ∑ n, (-1) ^ ↑n = if Even m then 0 else 1
• neg_one_geom_sum 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} [Ring R] {n : ℕ} : ∑ i ∈ Finset.range n, (-1) ^ i = if Even n then 0 else 1
• geom_sum_eq_zero_iff_neg_one 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} {n : ℕ} {x : R} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] (hn : n ≠ 0) : ∑ i ∈ Finset.range n, x ^ i = 0 ↔ x = -1 ∧ Even n
• geom_sum_neg_iff 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} {n : ℕ} {x : R} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] (hn : n ≠ 0) : ∑ i ∈ Finset.range n, x ^ i < 0 ↔ Even n ∧ x + 1 < 0
• geom_sum_alternating_of_le_neg_one 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} {x : R} [Ring R] [PartialOrder R] [IsOrderedRing R] (hx : x + 1 ≤ 0) (n : ℕ) : if Even n then ∑ i ∈ Finset.range n, x ^ i ≤ 0 else 1 ≤ ∑ i ∈ Finset.range n, x ^ i
• geom_sum_alternating_of_lt_neg_one 📋 Mathlib.Algebra.GeomSum
  {R : Type u_1} {n : ℕ} {x : R} [Ring R] [PartialOrder R] [IsStrictOrderedRing R] (hx : x + 1 < 0) (hn : 1 < n) : if Even n then ∑ i ∈ Finset.range n, x ^ i < 0 else 1 < ∑ i ∈ Finset.range n, x ^ i
• Finset.even_sum_iff_even_card_odd 📋 Mathlib.Algebra.BigOperators.Ring.Nat
  {ι : Type u_1} {s : Finset ι} (f : ι → ℕ) : Even (∑ i ∈ s, f i) ↔ Even {x ∈ s | Odd (f x)}.card
• Real.pow_arith_mean_le_arith_mean_pow_of_even 📋 Mathlib.Analysis.MeanInequalitiesPow
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) {n : ℕ} (hn : Even n) : (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n

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 b340a5b serving mathlib revision 846a90c