# Loogle!

## Result

Found 76 definitions mentioning tsum and LT.lt. Of these, 6 match your pattern(s).

- tsum_strict_mono Mathlib.Topology.Algebra.InfiniteSum.Order
`∀ {ι : Type u_1} {α : Type u_3} [inst : OrderedAddCommGroup α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalAddGroup α] [inst_3 : OrderClosedTopology α] {f g : ι → α}, Summable f → Summable g → f < g → ∑' (n : ι), f n < ∑' (n : ι), g n` - tsum_lt_tsum Mathlib.Topology.Algebra.InfiniteSum.Order
`∀ {ι : Type u_1} {α : Type u_3} [inst : OrderedAddCommGroup α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalAddGroup α] [inst_3 : OrderClosedTopology α] {f g : ι → α} {i : ι}, f ≤ g → f i < g i → Summable f → Summable g → ∑' (n : ι), f n < ∑' (n : ι), g n` - NNReal.tsum_strict_mono Mathlib.Topology.Instances.ENNReal
`∀ {α : Type u_1} {f g : α → NNReal}, Summable g → f < g → ∑' (n : α), f n < ∑' (n : α), g n` - NNReal.tsum_lt_tsum Mathlib.Topology.Instances.ENNReal
`∀ {α : Type u_1} {f g : α → NNReal} {i : α}, (∀ (a : α), f a ≤ g a) → f i < g i → Summable g → ∑' (n : α), f n < ∑' (n : α), g n` - ENNReal.tsum_lt_tsum Mathlib.Topology.Instances.ENNReal
`∀ {α : Type u_1} {f g : α → ENNReal} {i : α}, tsum f ≠ ⊤ → (∀ (a : α), f a ≤ g a) → f i < g i → ∑' (x : α), f x < ∑' (x : α), g x` - tsum_lt_tsum_of_nonneg Mathlib.Topology.Algebra.InfiniteSum.Real
`∀ {i : ℕ} {f g : ℕ → ℝ}, (∀ (b : ℕ), 0 ≤ f b) → (∀ (b : ℕ), f b ≤ g b) → f i < g i → Summable g → ∑' (n : ℕ), f n < ∑' (n : ℕ), g n`

## 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`