Loogle!

Result

Found 880 declarations whose name contains "zpow". Of these, 8 have a name containing "zpow" and "inj".

• zpow_left_injective 📋 Mathlib.Algebra.Group.Torsion
  {G : Type u_2} [Group G] [IsMulTorsionFree G] {n : ℤ} : n ≠ 0 → Function.Injective fun a => a ^ n
• zpow_left_inj 📋 Mathlib.Algebra.Group.Torsion
  {G : Type u_2} [Group G] [IsMulTorsionFree G] {n : ℤ} {a b : G} (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b
• zpow_left_injOn₀ 📋 Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
  {G₀ : Type u_3} [GroupWithZero G₀] [LinearOrder G₀] [PosMulStrictMono G₀] [MulPosMono G₀] {n : ℤ} : n ≠ 0 → Set.InjOn (fun a => a ^ n) {a | 0 ≤ a}
• zpow_right_injective₀ 📋 Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
  {G₀ : Type u_3} [GroupWithZero G₀] [LinearOrder G₀] {a : G₀} [PosMulStrictMono G₀] [ZeroLEOneClass G₀] (ha₀ : 0 < a) (ha₁ : a ≠ 1) : Function.Injective fun n => a ^ n
• zpow_right_inj₀ 📋 Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
  {G₀ : Type u_3} [GroupWithZero G₀] [LinearOrder G₀] {a : G₀} {m n : ℤ} [PosMulStrictMono G₀] [ZeroLEOneClass G₀] (ha₀ : 0 < a) (ha₁ : a ≠ 1) : a ^ m = a ^ n ↔ m = n
• zpow_left_inj₀ 📋 Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
  {G₀ : Type u_3} [GroupWithZero G₀] [LinearOrder G₀] {a b : G₀} {n : ℤ} [PosMulStrictMono G₀] [MulPosMono G₀] (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b
• zpow_right_inj 📋 Mathlib.Algebra.Order.Group.Basic
  {α : Type u_1} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] {a : α} (ha : 1 < a) {m n : ℤ} : a ^ m = a ^ n ↔ m = n
• injective_zpow_iff_not_isOfFinOrder 📋 Mathlib.GroupTheory.OrderOfElement
  {G : Type u_1} [Group G] {x : G} : (Function.Injective fun n => x ^ n) ↔ ¬IsOfFinOrder x

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.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

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 128218b serving mathlib revision 4644b1d