File: Mathlib/Topology/Algebra/Order/UpperLower.lean

full_name: HasUpperLowerClosure.isUpperSet_closure
code: /-- Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! -/
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
goal: IsUpperSet s → IsUpperSet (closure s)

full_name: HasUpperLowerClosure.isOpen_upperClosure
code: /-- Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! -/
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
goal: IsOpen s → IsOpen ↑(upperClosure s)

full_name: HasUpperLowerClosure.isOpen_lowerClosure
code: /-- Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! -/
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
goal: IsOpen s → IsOpen ↑(lowerClosure s)

full_name: HasUpperLowerClosure
code: /-- Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! -/
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
goal: Prop

full_name: HasUpperLowerClosure.isLowerSet_closure
code: /-- Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! -/
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
goal: IsLowerSet s → IsLowerSet (closure s)

full_name: OrderedAddCommGroup.to_hasUpperLowerClosure
code: @[to_additive]
instance (priority := 100) OrderedCommGroup.to_hasUpperLowerClosure [OrderedCommGroup α]
    [ContinuousConstSMul α α] : HasUpperLowerClosure α 
goal: HasUpperLowerClosure α

full_name: OrderedCommGroup.to_hasUpperLowerClosure
code: @[to_additive]
instance (priority := 100) OrderedCommGroup.to_hasUpperLowerClosure [OrderedCommGroup α]
    [ContinuousConstSMul α α] : HasUpperLowerClosure α 
goal: HasUpperLowerClosure α

full_name: IsUpperSet.closure
code: protected theorem IsUpperSet.closure : IsUpperSet s → IsUpperSet (closure s) 
goal: IsUpperSet s → IsUpperSet (closure s)

full_name: IsLowerSet.closure
code: protected theorem IsLowerSet.closure : IsLowerSet s → IsLowerSet (closure s) 
goal: IsLowerSet s → IsLowerSet (closure s)

full_name: IsOpen.upperClosure
code: protected theorem IsOpen.upperClosure : IsOpen s → IsOpen (upperClosure s : Set α) 
goal: IsOpen s → IsOpen ↑(upperClosure s)

full_name: IsOpen.lowerClosure
code: protected theorem IsOpen.lowerClosure : IsOpen s → IsOpen (lowerClosure s : Set α) 
goal: IsOpen s → IsOpen ↑(lowerClosure s)

full_name: instHasUpperLowerClosureOrderDual
code: instance : HasUpperLowerClosure αᵒᵈ 
goal: HasUpperLowerClosure αᵒᵈ

full_name: IsUpperSet.interior
code: protected theorem IsUpperSet.interior (h : IsUpperSet s) : IsUpperSet (interior s) 
goal: IsUpperSet (interior s)

full_name: IsLowerSet.interior
code: protected theorem IsLowerSet.interior (h : IsLowerSet s) : IsLowerSet (interior s) 
goal: IsLowerSet (interior s)

full_name: Set.OrdConnected.interior
code: protected theorem Set.OrdConnected.interior (h : s.OrdConnected) : (interior s).OrdConnected 
goal: (interior s).OrdConnected