Loogle!

Result

Found 17 declarations mentioning MonoidAlgebra.map.

• MonoidAlgebra.map_id 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} [Semiring R] (x : MonoidAlgebra R M) : MonoidAlgebra.map (AddMonoidHom.id R) x = x
• MonoidAlgebra.map 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x : MonoidAlgebra R M) : MonoidAlgebra S M
• MonoidAlgebra.map_sum 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {ι : Type u_1} {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (s : Finset ι) (x : ι → MonoidAlgebra R M) : MonoidAlgebra.map f (∑ i ∈ s, x i) = ∑ i ∈ s, MonoidAlgebra.map f (x i)
• MonoidAlgebra.map_zero 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) : MonoidAlgebra.map f 0 = 0
• MonoidAlgebra.map_single 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (r : R) (m : M) : MonoidAlgebra.map f (MonoidAlgebra.single m r) = MonoidAlgebra.single m (f r)
• MonoidAlgebra.map_map 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {M : Type u_6} [Semiring R] [Semiring S] [Semiring T] (f : S →+ T) (g : R →+ S) (x : MonoidAlgebra R M) : MonoidAlgebra.map f (MonoidAlgebra.map g x) = MonoidAlgebra.map (f.comp g) x
• MonoidAlgebra.map_add 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x y : MonoidAlgebra R M) : MonoidAlgebra.map f (x + y) = MonoidAlgebra.map f x + MonoidAlgebra.map f y
• MonoidAlgebra.map_one 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [Monoid M] (f : R →+* S) : MonoidAlgebra.map (↑f) 1 = 1
• MonoidAlgebra.map_neg 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Ring R] [Ring S] (f : R →+ S) (x : MonoidAlgebra R M) : MonoidAlgebra.map f (-x) = -MonoidAlgebra.map f x
• MonoidAlgebra.coe_mapRangeRingHom 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [Monoid M] (f : R →+* S) : ⇑(MonoidAlgebra.mapRingHom M f) = MonoidAlgebra.map ↑f
• MonoidAlgebra.coe_mapRingHom 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [Monoid M] (f : R →+* S) : ⇑(MonoidAlgebra.mapRingHom M f) = MonoidAlgebra.map ↑f
• MonoidAlgebra.map_apply 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x : MonoidAlgebra R M) (a✝ : M) : (MonoidAlgebra.map f x) a✝ = f (x.coeff a✝)
• MonoidAlgebra.map_sub 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Ring R] [Ring S] (f : R →+ S) (x y : MonoidAlgebra R M) : MonoidAlgebra.map f (x - y) = MonoidAlgebra.map f x - MonoidAlgebra.map f y
• MonoidAlgebra.ofCoeff_mapRange 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x : M →₀ R) : MonoidAlgebra.ofCoeff (Finsupp.mapRange ⇑f ⋯ x) = MonoidAlgebra.map f (MonoidAlgebra.ofCoeff x)
• MonoidAlgebra.coeff_map 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x : MonoidAlgebra R M) : (MonoidAlgebra.map f x).coeff = Finsupp.mapRange ⇑f ⋯ x.coeff
• MonoidAlgebra.map_support 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x : MonoidAlgebra R M) : (MonoidAlgebra.map f x).support = Finsupp.onFinset_support x.coeff.support (⇑f ∘ ⇑x.coeff)
• MonoidAlgebra.map_mul 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [Mul M] (f : R →+* S) (x y : MonoidAlgebra R M) : MonoidAlgebra.map (↑f) (x * y) = MonoidAlgebra.map (↑f) x * MonoidAlgebra.map (↑f) y

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. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 88c39f3 serving mathlib revision 9977002