Loogle!

Result

Found 12 declarations mentioning UpperSet.map.

• UpperSet.map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) : UpperSet α ≃o UpperSet β
• UpperSet.map_refl 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} [Preorder α] : UpperSet.map (OrderIso.refl α) = OrderIso.refl (UpperSet α)
• UpperSet.symm_map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) : (UpperSet.map f).symm = UpperSet.map f.symm
• UpperSet.coe_map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : UpperSet α) : ↑((UpperSet.map f) s) = ⇑f '' ↑s
• UpperSet.mem_map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α ≃o β} {s : UpperSet α} {b : β} : b ∈ (UpperSet.map f) s ↔ f.symm b ∈ s
• LowerSet.compl_map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : LowerSet α) : ((LowerSet.map f) s).compl = (UpperSet.map f) s.compl
• UpperSet.compl_map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : UpperSet α) : ((UpperSet.map f) s).compl = (LowerSet.map f) s.compl
• UpperSet.map_map 📋 Mathlib.Order.UpperLower.CompleteLattice
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] {s : UpperSet α} (g : β ≃o γ) (f : α ≃o β) : (UpperSet.map g) ((UpperSet.map f) s) = (UpperSet.map (f.trans g)) s
• UpperSet.map_Ici 📋 Mathlib.Order.UpperLower.Principal
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α) : (UpperSet.map f) (UpperSet.Ici a) = UpperSet.Ici (f a)
• UpperSet.map_Ioi 📋 Mathlib.Order.UpperLower.Principal
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α) : (UpperSet.map f) (UpperSet.Ioi a) = UpperSet.Ioi (f a)
• upperClosure_image 📋 Mathlib.Order.UpperLower.Closure
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} (f : α ≃o β) : upperClosure (⇑f '' s) = (UpperSet.map f) (upperClosure s)
• UpperSet.map.eq_1 📋 Mathlib.Order.UpperLower.Closure
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) : UpperSet.map f = { toFun := fun s => { carrier := ⇑f '' ↑s, upper' := ⋯ }, invFun := fun t => { carrier := ⇑f ⁻¹' ↑t, upper' := ⋯ }, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }

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