# Loogle!

## Result

Found 80 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} [inst : NonUnitalNonAssocSemiring α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalSemiring α] {f : ι → α} [inst_3 : T2Space α] (a : α), 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} [inst : DivisionSemiring α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalSemiring α] {f : ι → α} {a : α} [inst_3 : 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}, 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}, 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}, 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 witin the Lean4 VSCode language extension
using (by default) Ctrl-K Ctrl-S. You can also try 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:

By constant:

🔍`Real.sin`

finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:

🔍`"differ"`

finds all lemmas that have`"differ"`

somewhere in their lemma*name*.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`

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 `f46663a`

serving mathlib revision `c78db14`