Loogle!

Result

Found 20 declarations mentioning Lattice.toSemilatticeSup, CompleteLattice.toLattice, PartialOrder.toPreorder, Preorder.toLE, TopologicalSpace.instCompleteLattice, CompleteLattice.toBoundedOrder, TopologicalSpace, OrderTop.toTop, BoundedOrder.toOrderTop, SemilatticeSup.toPartialOrder, and Top.top. Of these, 20 match your pattern(s).

• instIndiscreteTopology 📋 Mathlib.Topology.Order
  {α : Type u_1} : IndiscreteTopology α
• inseparable_top 📋 Mathlib.Topology.Order
  {α : Type u} (x y : α) : Inseparable x y
• IndiscreteTopology.eq_top 📋 Mathlib.Topology.Order
  (α : Type u_2) {inst✝ : TopologicalSpace α} [self : IndiscreteTopology α] : inst✝ = ⊤
• IndiscreteTopology.mk 📋 Mathlib.Topology.Order
  {α : Type u_2} [TopologicalSpace α] (eq_top : inst✝ = ⊤) : IndiscreteTopology α
• NontrivialTopology.mk 📋 Mathlib.Topology.Order
  {α : Type u_2} [TopologicalSpace α] (ne_top : inst✝ ≠ ⊤) : NontrivialTopology α
• NontrivialTopology.ne_top 📋 Mathlib.Topology.Order
  (α : Type u_2) {inst✝ : TopologicalSpace α} [self : NontrivialTopology α] : inst✝ ≠ ⊤
• continuous_top 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace α} : Continuous f
• indiscreteTopology_iff 📋 Mathlib.Topology.Order
  (α : Type u_2) [TopologicalSpace α] : IndiscreteTopology α ↔ inst✝ = ⊤
• nontrivialTopology_iff 📋 Mathlib.Topology.Order
  (α : Type u_2) [TopologicalSpace α] : NontrivialTopology α ↔ inst✝ ≠ ⊤
• nhds_top 📋 Mathlib.Topology.Order
  {α : Type u} {a : α} : nhds a = ⊤
• induced_const 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace α] {x : α} : TopologicalSpace.induced (fun x_1 => x) t = ⊤
• TopologicalSpace.eq_top_iff_forall_inseparable 📋 Mathlib.Topology.Order
  {α : Type u} {t : TopologicalSpace α} : t = ⊤ ↔ ∀ (x y : α), Inseparable x y
• TopologicalSpace.ne_top_iff_exists_not_inseparable 📋 Mathlib.Topology.Order
  {α : Type u} {t : TopologicalSpace α} : t ≠ ⊤ ↔ ∃ x y, ¬Inseparable x y
• TopologicalSpace.isOpen_top_iff 📋 Mathlib.Topology.Order
  {α : Type u_2} (U : Set α) : IsOpen U ↔ U = ∅ ∨ U = Set.univ
• induced_top 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} {g : β → α} : TopologicalSpace.induced g ⊤ = ⊤
• UniformSpace.toTopologicalSpace_top 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : ⊤.toTopologicalSpace = ⊤
• AddGroupTopology.toTopologicalSpace_top 📋 Mathlib.Topology.Algebra.Group.GroupTopology
  {α : Type u} [AddGroup α] : ⊤.toTopologicalSpace = ⊤
• GroupTopology.toTopologicalSpace_top 📋 Mathlib.Topology.Algebra.Group.GroupTopology
  {α : Type u} [Group α] : ⊤.toTopologicalSpace = ⊤
• TopCat.Presheaf.isSheaf_of_isTerminal_of_indiscrete 📋 Mathlib.Topology.Sheaves.PUnit
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (hind : X.str = ⊤) (F : TopCat.Presheaf C X) (it : CategoryTheory.Limits.IsTerminal (F.obj (Opposite.op ⊥))) : F.IsSheaf
• TopCat.Presheaf.isSheaf_iff_isTerminal_of_indiscrete 📋 Mathlib.Topology.Sheaves.PUnit
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (hind : X.str = ⊤) (F : TopCat.Presheaf C X) : F.IsSheaf ↔ Nonempty (CategoryTheory.Limits.IsTerminal (F.obj (Opposite.op ⊥)))

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 401c76f serving mathlib revision a3d2529