Loogle!

Result

Found 15 declarations mentioning SProd.

• SProd 📋 Mathlib.Data.SProd
  (α : Type u) (β : Type v) (γ : outParam (Type w)) : Type (max (max u v) w)
• SProd.mk 📋 Mathlib.Data.SProd
  {α : Type u} {β : Type v} {γ : outParam (Type w)} (sprod : α → β → γ) : SProd α β γ
• SProd.sprod 📋 Mathlib.Data.SProd
  {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : SProd α β γ] : α → β → γ
• Set.instSProd 📋 Mathlib.Data.Set.Operations
  {α : Type u} {β : Type v} : SProd (Set α) (Set β) (Set (α × β))
• List.instSProd 📋 Mathlib.Data.List.Defs
  {α : Type u_1} {β : Type u_2} : SProd (List α) (List β) (List (α × β))
• Multiset.instSProd 📋 Mathlib.Data.Multiset.Bind
  {α : Type u_1} {β : Type v} : SProd (Multiset α) (Multiset β) (Multiset (α × β))
• Finset.instSProd 📋 Mathlib.Data.Finset.Prod
  {α : Type u_1} {β : Type u_2} : SProd (Finset α) (Finset β) (Finset (α × β))
• Filter.instSProd 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} {β : Type u_2} : SProd (Filter α) (Filter β) (Filter (α × β))
• TopologicalSpace.Clopens.instSProdProd 📋 Mathlib.Topology.Sets.Closeds
  {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] : SProd (TopologicalSpace.Clopens α) (TopologicalSpace.Clopens β) (TopologicalSpace.Clopens (α × β))
• TopologicalSpace.CompactOpens.instSProdProd 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : SProd (TopologicalSpace.CompactOpens α) (TopologicalSpace.CompactOpens β) (TopologicalSpace.CompactOpens (α × β))
• TopologicalSpace.Compacts.instSProdProd 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : SProd (TopologicalSpace.Compacts α) (TopologicalSpace.Compacts β) (TopologicalSpace.Compacts (α × β))
• TopologicalSpace.NonemptyCompacts.instSProdProd 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : SProd (TopologicalSpace.NonemptyCompacts α) (TopologicalSpace.NonemptyCompacts β) (TopologicalSpace.NonemptyCompacts (α × β))
• TopologicalSpace.PositiveCompacts.instSProdProd 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : SProd (TopologicalSpace.PositiveCompacts α) (TopologicalSpace.PositiveCompacts β) (TopologicalSpace.PositiveCompacts (α × β))
• LowerSet.instSProd 📋 Mathlib.Order.UpperLower.Prod
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : SProd (LowerSet α) (LowerSet β) (LowerSet (α × β))
• UpperSet.instSProd 📋 Mathlib.Order.UpperLower.Prod
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : SProd (UpperSet α) (UpperSet β) (UpperSet (α × β))

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 6ff4759 serving mathlib revision 519f454