Loogle!

Result

Found 4 declarations mentioning Module.DirectLimit.map.

• Module.DirectLimit.map_id 📋 Mathlib.Algebra.Colimit.Module
  {R : Type u_1} [Semiring R] {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [DecidableEq ι] [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} : Module.DirectLimit.map (fun x => LinearMap.id) ⋯ = LinearMap.id
• Module.DirectLimit.map 📋 Mathlib.Algebra.Colimit.Module
  {R : Type u_1} [Semiring R] {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [DecidableEq ι] [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} (g : (i : ι) → G i →ₗ[R] G' i) (hg : ∀ (i j : ι) (h : i ≤ j), g j ∘ₗ f i j h = f' i j h ∘ₗ g i) : Module.DirectLimit G f →ₗ[R] Module.DirectLimit G' f'
• Module.DirectLimit.map_apply_of 📋 Mathlib.Algebra.Colimit.Module
  {R : Type u_1} [Semiring R] {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [DecidableEq ι] [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} (g : (i : ι) → G i →ₗ[R] G' i) (hg : ∀ (i j : ι) (h : i ≤ j), g j ∘ₗ f i j h = f' i j h ∘ₗ g i) {i : ι} (x : G i) : (Module.DirectLimit.map g hg) ((Module.DirectLimit.of R ι G f i) x) = (Module.DirectLimit.of R ι G' f' i) ((g i) x)
• Module.DirectLimit.map_comp 📋 Mathlib.Algebra.Colimit.Module
  {R : Type u_1} [Semiring R] {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [DecidableEq ι] [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} {G' : ι → Type u_5} [(i : ι) → AddCommMonoid (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} {G'' : ι → Type u_6} [(i : ι) → AddCommMonoid (G'' i)] [(i : ι) → Module R (G'' i)] {f'' : (i j : ι) → i ≤ j → G'' i →ₗ[R] G'' j} (g₁ : (i : ι) → G i →ₗ[R] G' i) (g₂ : (i : ι) → G' i →ₗ[R] G'' i) (hg₁ : ∀ (i j : ι) (h : i ≤ j), g₁ j ∘ₗ f i j h = f' i j h ∘ₗ g₁ i) (hg₂ : ∀ (i j : ι) (h : i ≤ j), g₂ j ∘ₗ f' i j h = f'' i j h ∘ₗ g₂ i) : Module.DirectLimit.map g₂ hg₂ ∘ₗ Module.DirectLimit.map g₁ hg₁ = Module.DirectLimit.map (fun i => g₂ i ∘ₗ 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