Loogle!

Result

Found 130 declarations mentioning AddMonoid.toAddZeroClass, Nat.instAddMonoid, Nat and AddMonoidHom. Of these, 23 match your pattern(s).

• multiplesHom 📋 Mathlib.Algebra.Group.Nat.Hom
  (M : Type u_1) [AddMonoid M] : M ≃ (ℕ →+ M)
• multiplesAddHom 📋 Mathlib.Algebra.Group.Nat.Hom
  (M : Type u_1) [AddCommMonoid M] : M ≃+ (ℕ →+ M)
• multiplesAddHom.eq_1 📋 Mathlib.Algebra.Group.Nat.Hom
  (M : Type u_1) [AddCommMonoid M] : multiplesAddHom M = { toEquiv := multiplesHom M, map_add' := ⋯ }
• AddMonoidHom.ext_nat 📋 Mathlib.Algebra.Group.Nat.Hom
  {A : Type u_2} [AddZeroClass A] {f g : ℕ →+ A} : f 1 = g 1 → f = g
• AddMonoidHom.apply_nat 📋 Mathlib.Algebra.Group.Nat.Hom
  {M : Type u_1} [AddMonoid M] (f : ℕ →+ M) (n : ℕ) : f n = n • f 1
• AddMonoidHom.ext_nat_iff 📋 Mathlib.Algebra.Group.Nat.Hom
  {A : Type u_2} [AddZeroClass A] {f g : ℕ →+ A} : f = g ↔ f 1 = g 1
• multiplesHom_apply 📋 Mathlib.Algebra.Group.Nat.Hom
  {M : Type u_1} [AddMonoid M] (x : M) (n : ℕ) : ((multiplesHom M) x) n = n • x
• powersHom.eq_1 📋 Mathlib.Algebra.Group.Nat.Hom
  (M : Type u_1) [Monoid M] : powersHom M = Additive.ofMul.trans ((multiplesHom (Additive M)).trans AddMonoidHom.toMultiplicative'')
• multiplesHom_symm_apply 📋 Mathlib.Algebra.Group.Nat.Hom
  {M : Type u_1} [AddMonoid M] (f : ℕ →+ M) : (multiplesHom M).symm f = f 1
• multiplesHom.eq_1 📋 Mathlib.Algebra.Group.Nat.Hom
  (M : Type u_1) [AddMonoid M] : multiplesHom M = { toFun := fun x => { toFun := fun n => n • x, map_zero' := ⋯, map_add' := ⋯ }, invFun := fun f => f 1, left_inv := ⋯, right_inv := ⋯ }
• multiplesAddHom_apply 📋 Mathlib.Algebra.Group.Nat.Hom
  {M : Type u_1} [AddCommMonoid M] (x : M) (n : ℕ) : ((multiplesAddHom M) x) n = n • x
• multiplesAddHom_symm_apply 📋 Mathlib.Algebra.Group.Nat.Hom
  {M : Type u_1} [AddCommMonoid M] (f : ℕ →+ M) : (multiplesAddHom M).symm f = f 1
• Nat.castAddMonoidHom 📋 Mathlib.Data.Nat.Cast.Basic
  (α : Type u_3) [AddMonoidWithOne α] : ℕ →+ α
• Nat.coe_castAddMonoidHom 📋 Mathlib.Data.Nat.Cast.Basic
  {α : Type u_1} [AddMonoidWithOne α] : ⇑(Nat.castAddMonoidHom α) = Nat.cast
• Multiset.replicateAddMonoidHom 📋 Mathlib.Algebra.Order.Group.Multiset
  {α : Type u_1} (a : α) : ℕ →+ Multiset α
• Multiset.replicateAddMonoidHom_apply 📋 Mathlib.Algebra.Order.Group.Multiset
  {α : Type u_1} (a : α) (n : ℕ) : (Multiset.replicateAddMonoidHom a) n = Multiset.replicate n a
• AddSubmonoid.closure_singleton_eq 📋 Mathlib.Algebra.Group.Submonoid.Membership
  {A : Type u_2} [AddMonoid A] (x : A) : AddSubmonoid.closure {x} = AddMonoidHom.mrange ((multiplesHom A) x)
• AddSubmonoid.multiples.eq_1 📋 Mathlib.Algebra.Group.Submonoid.Membership
  {A : Type u_2} [AddMonoid A] (x : A) : AddSubmonoid.multiples x = (AddMonoidHom.mrange ((multiplesHom A) x)).copy (Set.range fun i => i • x) ⋯
• AddSubmonoid.one_eq_mrange 📋 Mathlib.Algebra.Ring.Submonoid.Pointwise
  {R : Type u_2} [AddMonoidWithOne R] : 1 = AddMonoidHom.mrange (Nat.castAddMonoidHom R)
• AddMonoidHom.ENatMap 📋 Mathlib.Data.ENat.Basic
  {N : Type u_2} [AddZeroClass N] (f : ℕ →+ N) : ℕ∞ →+ WithTop N
• AddMonoidHom.ENatMap_apply 📋 Mathlib.Data.ENat.Basic
  {N : Type u_2} [AddZeroClass N] (f : ℕ →+ N) : ⇑f.ENatMap = ENat.map ⇑f
• AddMonoidHom.fromNatEquiv 📋 Mathlib.Algebra.Category.MonCat.ForgetCorepresentable
  (α : Type u) [AddMonoid α] : (ℕ →+ α) ≃ α
• uliftMultiplesHom.eq_1 📋 Mathlib.Algebra.Category.MonCat.ForgetCorepresentable
  (M : Type u) [AddMonoid M] : uliftMultiplesHom M = (multiplesHom M).trans (AddMonoidHom.precompEquiv AddEquiv.ulift M)

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 19971e9 serving mathlib revision bce1d65