Loogle!

Result

Found 19 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, 19 match your pattern(s).

• inseparable_top 📋 Mathlib.Topology.Order
  {α : Type u} (x y : α) : Inseparable x y
• continuous_top 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} {f : α → β} {t : TopologicalSpace α} : Continuous f
• 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 ⊤ = ⊤
• SeparationQuotient.nontrivial_iff 📋 Mathlib.Topology.Inseparable
  {α : Type u_4} {t : TopologicalSpace α} : Nontrivial (SeparationQuotient α) ↔ t ≠ ⊤
• SeparationQuotient.subsingleton_iff 📋 Mathlib.Topology.Inseparable
  {α : Type u_4} {t : TopologicalSpace α} : Subsingleton (SeparationQuotient α) ↔ t = ⊤
• SeparationQuotient.inseparableSetoid_eq_top_iff 📋 Mathlib.Topology.Inseparable
  {α : Type u_4} {t : TopologicalSpace α} : inseparableSetoid α = ⊤ ↔ t = ⊤
• 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 = ⊤
• AddGroupTopology.instTop.eq_1 📋 Mathlib.Topology.Algebra.Group.GroupTopology
  {α : Type u} [AddGroup α] : AddGroupTopology.instTop = { top := { toTopologicalSpace := ⊤, toIsTopologicalAddGroup := ⋯ } }
• GroupTopology.instTop.eq_1 📋 Mathlib.Topology.Algebra.Group.GroupTopology
  {α : Type u} [Group α] : GroupTopology.instTop = { top := { toTopologicalSpace := ⊤, toIsTopologicalGroup := ⋯ } }
• UniformSpace.toTopologicalSpace_top 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} : ⊤.toTopologicalSpace = ⊤
• TopologicalSpace.Opens.eq_bot_or_top 📋 Mathlib.Topology.Sets.Opens
  {α : Type u_5} [t : TopologicalSpace α] (h : t = ⊤) (U : TopologicalSpace.Opens α) : U = ⊥ ∨ U = ⊤
• 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 15beac7 serving mathlib revision 1827e11