Loogle!

Result

Found 5 declarations mentioning AddCommGroup.DirectLimit.map.

• AddCommGroup.DirectLimit.map_id 📋 Mathlib.Algebra.Colimit.Module
  {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [DecidableEq ι] : AddCommGroup.DirectLimit.map (fun x => AddMonoidHom.id (G x)) ⋯ = AddMonoidHom.id (AddCommGroup.DirectLimit G f)
• AddCommGroup.DirectLimit.map 📋 Mathlib.Algebra.Colimit.Module
  {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [DecidableEq ι] {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} (g : (i : ι) → G i →+ G' i) (hg : ∀ (i j : ι) (h : i ≤ j), (g j).comp (f i j h) = (f' i j h).comp (g i)) : AddCommGroup.DirectLimit G f →+ AddCommGroup.DirectLimit G' f'
• AddCommGroup.DirectLimit.map_apply_of 📋 Mathlib.Algebra.Colimit.Module
  {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [DecidableEq ι] {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} (g : (i : ι) → G i →+ G' i) (hg : ∀ (i j : ι) (h : i ≤ j), (g j).comp (f i j h) = (f' i j h).comp (g i)) {i : ι} (x : G i) : (AddCommGroup.DirectLimit.map g hg) ((AddCommGroup.DirectLimit.of G f i) x) = (AddCommGroup.DirectLimit.of G' f' i) ((g i) x)
• AddCommGroup.DirectLimit.congr.eq_1 📋 Mathlib.Algebra.Colimit.Module
  {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [DecidableEq ι] {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} (e : (i : ι) → G i ≃+ G' i) (he : ∀ (i j : ι) (h : i ≤ j), (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp ↑(e i)) : AddCommGroup.DirectLimit.congr e he = (AddCommGroup.DirectLimit.map (fun x => ↑(e x)) he).toAddEquiv (AddCommGroup.DirectLimit.map (fun i => ↑(e i).symm) ⋯) ⋯ ⋯
• AddCommGroup.DirectLimit.map_comp 📋 Mathlib.Algebra.Colimit.Module
  {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [DecidableEq ι] {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} {G'' : ι → Type u_6} [(i : ι) → AddCommMonoid (G'' i)] {f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j} (g₁ : (i : ι) → G i →+ G' i) (g₂ : (i : ι) → G' i →+ G'' i) (hg₁ : ∀ (i j : ι) (h : i ≤ j), (g₁ j).comp (f i j h) = (f' i j h).comp (g₁ i)) (hg₂ : ∀ (i j : ι) (h : i ≤ j), (g₂ j).comp (f' i j h) = (f'' i j h).comp (g₂ i)) : (AddCommGroup.DirectLimit.map g₂ hg₂).comp (AddCommGroup.DirectLimit.map g₁ hg₁) = AddCommGroup.DirectLimit.map (fun i => (g₂ i).comp (g₁ i)) ⋯

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