Loogle!

Result

Found 16 declarations mentioning Monoid.Coprod.map.

• Monoid.Coprod.map 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') : Monoid.Coprod M N →* Monoid.Coprod M' N'
• Monoid.Coprod.map_id_id 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] : Monoid.Coprod.map (MonoidHom.id M) (MonoidHom.id N) = MonoidHom.id (Monoid.Coprod M N)
• Monoid.Coprod.map_comp_inl 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') : (Monoid.Coprod.map f g).comp Monoid.Coprod.inl = Monoid.Coprod.inl.comp f
• Monoid.Coprod.map_comp_inr 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') : (Monoid.Coprod.map f g).comp Monoid.Coprod.inr = Monoid.Coprod.inr.comp g
• Monoid.Coprod.swap_comp_map 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') : (Monoid.Coprod.swap M' N').comp (Monoid.Coprod.map f g) = (Monoid.Coprod.map g f).comp (Monoid.Coprod.swap M N)
• Monoid.Coprod.map_comp_map 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] {M'' : Type u_6} {N'' : Type u_7} [MulOneClass M''] [MulOneClass N''] (f' : M' →* M'') (g' : N' →* N'') (f : M →* M') (g : N →* N') : (Monoid.Coprod.map f' g').comp (Monoid.Coprod.map f g) = Monoid.Coprod.map (f'.comp f) (g'.comp g)
• Monoid.Coprod.map_apply_inl 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') (x : M) : (Monoid.Coprod.map f g) (Monoid.Coprod.inl x) = Monoid.Coprod.inl (f x)
• Monoid.Coprod.map_apply_inr 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') (x : N) : (Monoid.Coprod.map f g) (Monoid.Coprod.inr x) = Monoid.Coprod.inr (g x)
• Monoid.Coprod.map.eq_1 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') : Monoid.Coprod.map f g = Monoid.Coprod.clift (Monoid.Coprod.mk.comp (FreeMonoid.map (Sum.map ⇑f ⇑g))) ⋯ ⋯ ⋯ ⋯
• Monoid.Coprod.map_map 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] {M'' : Type u_6} {N'' : Type u_7} [MulOneClass M''] [MulOneClass N''] (f' : M' →* M'') (g' : N' →* N'') (f : M →* M') (g : N →* N') (x : Monoid.Coprod M N) : (Monoid.Coprod.map f' g') ((Monoid.Coprod.map f g) x) = (Monoid.Coprod.map (f'.comp f) (g'.comp g)) x
• Monoid.Coprod.swap_map 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') (x : Monoid.Coprod M N) : (Monoid.Coprod.swap M' N') ((Monoid.Coprod.map f g) x) = (Monoid.Coprod.map g f) ((Monoid.Coprod.swap M N) x)
• MulEquiv.coprodCongr.eq_1 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (e : M ≃* N) (e' : M' ≃* N') : e.coprodCongr e' = (Monoid.Coprod.map ↑e ↑e').toMulEquiv (Monoid.Coprod.map ↑e.symm ↑e'.symm) ⋯ ⋯
• MulEquiv.coprodCongr_apply 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (e : M ≃* N) (e' : M' ≃* N') : ⇑(e.coprodCongr e') = ⇑(Monoid.Coprod.map ↑e ↑e')
• MulEquiv.coprodCongr_symm_apply 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (e : M ≃* N) (e' : M' ≃* N') : ⇑(e.coprodCongr e').symm = ⇑(Monoid.Coprod.map ↑e.symm ↑e'.symm)
• Monoid.Coprod.map_mk_ofList 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (f : M →* M') (g : N →* N') (l : List (M ⊕ N)) : (Monoid.Coprod.map f g) (Monoid.Coprod.mk (FreeMonoid.ofList l)) = Monoid.Coprod.mk (FreeMonoid.ofList (List.map (Sum.map ⇑f ⇑g) l))
• MulEquiv.coprodAssoc.eq_1 📋 Mathlib.GroupTheory.Coprod.Basic
  (M : Type u_1) (N : Type u_2) (P : Type u_3) [Monoid M] [Monoid N] [Monoid P] : MulEquiv.coprodAssoc M N P = (Monoid.Coprod.lift (Monoid.Coprod.map (MonoidHom.id M) Monoid.Coprod.inl) (Monoid.Coprod.inr.comp Monoid.Coprod.inr)).toMulEquiv (Monoid.Coprod.lift (Monoid.Coprod.inl.comp Monoid.Coprod.inl) (Monoid.Coprod.map Monoid.Coprod.inr (MonoidHom.id P))) ⋯ ⋯

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 ff04530 serving mathlib revision 8623f65