Loogle!
Result
Found 5 declarations mentioning Ring.DirectLimit.map.
- Ring.DirectLimit.map 📋 Mathlib.Algebra.Colimit.Ring
{ι : Type u_1} [Preorder ι] {G : ι → Type u_2} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type u_4} [(i : ι) → CommRing (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)) : (Ring.DirectLimit G fun x x_1 h => ⇑(f x x_1 h)) →+* Ring.DirectLimit G' fun x x_1 h => ⇑(f' x x_1 h) - Ring.DirectLimit.map_id 📋 Mathlib.Algebra.Colimit.Ring
{ι : Type u_1} [Preorder ι] {G : ι → Type u_2} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} : Ring.DirectLimit.map (fun x => RingHom.id (G x)) ⋯ = RingHom.id (Ring.DirectLimit G fun x x_1 h => ⇑(f x x_1 h)) - Ring.DirectLimit.map_apply_of 📋 Mathlib.Algebra.Colimit.Ring
{ι : Type u_1} [Preorder ι] {G : ι → Type u_2} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type u_4} [(i : ι) → CommRing (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) : (Ring.DirectLimit.map g hg) ((Ring.DirectLimit.of G (fun x x_1 h => ⇑(f x x_1 h)) i) x) = (Ring.DirectLimit.of G' (fun x x_1 h => ⇑(f' x x_1 h)) i) ((g i) x) - Ring.DirectLimit.congr.eq_1 📋 Mathlib.Algebra.Colimit.Ring
{ι : Type u_1} [Preorder ι] {G : ι → Type u_2} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type u_4} [(i : ι) → CommRing (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).toRingHom.comp (f i j h) = (f' i j h).comp ↑(e i)) : Ring.DirectLimit.congr e he = RingEquiv.ofRingHom (Ring.DirectLimit.map (fun x => ↑(e x)) he) (Ring.DirectLimit.map (fun i => ↑(e i).symm) ⋯) ⋯ ⋯ - Ring.DirectLimit.map_comp 📋 Mathlib.Algebra.Colimit.Ring
{ι : Type u_1} [Preorder ι] {G : ι → Type u_2} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type u_4} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} {G'' : ι → Type u_5} [(i : ι) → CommRing (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)) : (Ring.DirectLimit.map g₂ hg₂).comp (Ring.DirectLimit.map g₁ hg₁) = Ring.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:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.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
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 findtsum_lt_tsum
even though the hypothesisf 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