Loogle!

Result

Found 82 definitions mentioning HMul.hMul, tsum and Eq. Of these, 7 match your pattern(s).

• Summable.tsum_mul_left 📋 Mathlib.Topology.Algebra.InfiniteSum.Ring
  {ι : Type u_1} {α : Type u_4} [NonUnitalNonAssocSemiring α] [TopologicalSpace α] [TopologicalSemiring α] {f : ι → α} [T2Space α] (a : α) (hf : Summable f) : ∑' (i : ι), a * f i = a * ∑' (i : ι), f i
• tsum_mul_left 📋 Mathlib.Topology.Algebra.InfiniteSum.Ring
  {ι : Type u_1} {α : Type u_4} [DivisionSemiring α] [TopologicalSpace α] [TopologicalSemiring α] {f : ι → α} {a : α} [T2Space α] : ∑' (x : ι), a * f x = a * ∑' (x : ι), f x
• NNReal.tsum_mul_left 📋 Mathlib.Topology.Instances.NNReal
  {α : Type u_1} (a : NNReal) (f : α → NNReal) : ∑' (x : α), a * f x = a * ∑' (x : α), f x
• ENNReal.tsum_mul_left 📋 Mathlib.Topology.Instances.ENNReal
  {α : Type u_1} {a : ENNReal} {f : α → ENNReal} : ∑' (i : α), a * f i = a * ∑' (i : α), f i
• Real.tsum_exp_neg_mul_int_sq 📋 Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation
  {a : ℝ} (ha : 0 < a) : ∑' (n : ℤ), Real.exp (-Real.pi * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), Real.exp (-Real.pi / a * ↑n ^ 2)
• Complex.tsum_exp_neg_mul_int_sq 📋 Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation
  {a : ℂ} (ha : 0 < a.re) : ∑' (n : ℤ), Complex.exp (-↑Real.pi * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), Complex.exp (-↑Real.pi / a * ↑n ^ 2)
• Complex.tsum_exp_neg_quadratic 📋 Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation
  {a : ℂ} (ha : 0 < a.re) (b : ℂ) : ∑' (n : ℤ), Complex.exp (-↑Real.pi * a * ↑n ^ 2 + 2 * ↑Real.pi * b * ↑n) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), Complex.exp (-↑Real.pi / a * (↑n + Complex.I * b) ^ 2)

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, _ * _, _ ^ _, |- _ < _ → _
woould 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 4e1aab0 serving mathlib revision 79ca167