Loogle!

Result

Found 8 declarations mentioning PolynomialModule.map.

• PolynomialModule.map 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (R' : Type u_4) {M' : Type u_5} [CommRing R'] [AddCommGroup M'] [Module R' M'] [Module R M'] (f : M →ₗ[R] M') : PolynomialModule R M →ₗ[R] PolynomialModule R' M'
• PolynomialModule.map_single 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (R' : Type u_4) {M' : Type u_5} [CommRing R'] [AddCommGroup M'] [Module R' M'] [Module R M'] (f : M →ₗ[R] M') (i : ℕ) (m : M) : (PolynomialModule.map R' f) ((PolynomialModule.single R i) m) = (PolynomialModule.single R' i) (f m)
• PolynomialModule.eval_map' 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (f : M →ₗ[R] M) (q : PolynomialModule R M) (r : R) : (PolynomialModule.eval r) ((PolynomialModule.map R f) q) = f ((PolynomialModule.eval r) q)
• PolynomialModule.eval_map 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (R' : Type u_4) {M' : Type u_5} [CommRing R'] [AddCommGroup M'] [Module R' M'] [Module R M'] [Algebra R R'] [IsScalarTower R R' M'] (f : M →ₗ[R] M') (q : PolynomialModule R M) (r : R) : (PolynomialModule.eval ((algebraMap R R') r)) ((PolynomialModule.map R' f) q) = f ((PolynomialModule.eval r) q)
• PolynomialModule.map_smul 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (R' : Type u_4) {M' : Type u_5} [CommRing R'] [AddCommGroup M'] [Module R' M'] [Module R M'] [Algebra R R'] [IsScalarTower R R' M'] (f : M →ₗ[R] M') (p : Polynomial R) (q : PolynomialModule R M) : (PolynomialModule.map R' f) (p • q) = Polynomial.map (algebraMap R R') p • (PolynomialModule.map R' f) q
• PolynomialModule.comp_apply 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (p : Polynomial R) (a✝ : PolynomialModule R M) : (PolynomialModule.comp p) a✝ = (PolynomialModule.eval p) ((PolynomialModule.map (Polynomial R) (PolynomialModule.lsingle R 0)) a✝)
• PolynomialModule.aeval_equivPolynomial 📋 Mathlib.Algebra.Polynomial.Module.Basic
  {R : Type u_1} [CommRing R] {S : Type u_6} [CommRing S] [Algebra S R] (f : PolynomialModule S S) (x : R) : (Polynomial.aeval x) (PolynomialModule.equivPolynomial f) = (PolynomialModule.eval x) ((PolynomialModule.map R (Algebra.linearMap S R)) f)
• Derivation.apply_aeval_eq' 📋 Mathlib.RingTheory.Derivation.MapCoeffs
  {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommRing R] [CommRing A] [Algebra R A] [AddCommGroup M] [Module A M] [Module R M] (d : Derivation R A M) {B : Type u_4} {M' : Type u_5} [CommRing B] [Algebra R B] [Algebra A B] [AddCommGroup M'] [Module B M'] [Module R M'] [Module A M'] (d' : Derivation R B M') (f : M →ₗ[A] M') (h : ∀ (a : A), f (d a) = d' ((algebraMap A B) a)) (x : B) (p : Polynomial A) : d' ((Polynomial.aeval x) p) = (PolynomialModule.eval x) ((PolynomialModule.map B f) (d.mapCoeffs p)) + (Polynomial.aeval x) (Polynomial.derivative p) • d' 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.

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