File: Mathlib/Analysis/Normed/Order/UpperLower.lean

full_name: IsUpperSet.thickening'
code: @[to_additive IsUpperSet.thickening]
protected theorem IsUpperSet.thickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (thickening ε s) 
goal: IsUpperSet (Metric.thickening ε s)

full_name: IsUpperSet.thickening
code: @[to_additive IsUpperSet.thickening]
protected theorem IsUpperSet.thickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (thickening ε s) 
goal: IsUpperSet (Metric.thickening ε s)

full_name: IsLowerSet.thickening'
code: @[to_additive IsLowerSet.thickening]
protected theorem IsLowerSet.thickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (thickening ε s) 
goal: IsLowerSet (Metric.thickening ε s)

full_name: IsLowerSet.thickening
code: @[to_additive IsLowerSet.thickening]
protected theorem IsLowerSet.thickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (thickening ε s) 
goal: IsLowerSet (Metric.thickening ε s)

full_name: IsUpperSet.cthickening
code: @[to_additive IsUpperSet.cthickening]
protected theorem IsUpperSet.cthickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (cthickening ε s) 
goal: IsUpperSet (Metric.cthickening ε s)

full_name: IsUpperSet.cthickening'
code: @[to_additive IsUpperSet.cthickening]
protected theorem IsUpperSet.cthickening' (hs : IsUpperSet s) (ε : ℝ) :
    IsUpperSet (cthickening ε s) 
goal: IsUpperSet (Metric.cthickening ε s)

full_name: IsLowerSet.cthickening
code: @[to_additive IsLowerSet.cthickening]
protected theorem IsLowerSet.cthickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (cthickening ε s) 
goal: IsLowerSet (Metric.cthickening ε s)

full_name: IsLowerSet.cthickening'
code: @[to_additive IsLowerSet.cthickening]
protected theorem IsLowerSet.cthickening' (hs : IsLowerSet s) (ε : ℝ) :
    IsLowerSet (cthickening ε s) 
goal: IsLowerSet (Metric.cthickening ε s)

full_name: upperClosure_interior_subset
code: @[to_additive upperClosure_interior_subset] lemma upperClosure_interior_subset' (s : Set α) :
    (upperClosure (interior s) : Set α) ⊆ interior (upperClosure s) 
goal: ↑(upperClosure (interior s)) ⊆ interior ↑(upperClosure s)

full_name: upperClosure_interior_subset'
code: @[to_additive upperClosure_interior_subset] lemma upperClosure_interior_subset' (s : Set α) :
    (upperClosure (interior s) : Set α) ⊆ interior (upperClosure s) 
goal: ↑(upperClosure (interior s)) ⊆ interior ↑(upperClosure s)

full_name: lowerClosure_interior_subset
code: @[to_additive lowerClosure_interior_subset] lemma lowerClosure_interior_subset' (s : Set α) :
    (lowerClosure (interior s) : Set α) ⊆ interior (lowerClosure s) 
goal: ↑(lowerClosure (interior s)) ⊆ interior ↑(lowerClosure s)

full_name: lowerClosure_interior_subset'
code: @[to_additive lowerClosure_interior_subset] lemma lowerClosure_interior_subset' (s : Set α) :
    (lowerClosure (interior s) : Set α) ⊆ interior (lowerClosure s) 
goal: ↑(lowerClosure (interior s)) ⊆ interior ↑(lowerClosure s)

full_name: IsUpperSet.mem_interior_of_forall_lt
code: theorem IsUpperSet.mem_interior_of_forall_lt (hs : IsUpperSet s) (hx : x ∈ closure s)
    (h : ∀ i, x i < y i) : y ∈ interior s 
goal: y ∈ interior s

full_name: IsLowerSet.mem_interior_of_forall_lt
code: theorem IsLowerSet.mem_interior_of_forall_lt (hs : IsLowerSet s) (hx : x ∈ closure s)
    (h : ∀ i, y i < x i) : y ∈ interior s 
goal: y ∈ interior s

full_name: dist_inf_sup_pi
code: lemma dist_inf_sup_pi (x y : ι → ℝ) : dist (x ⊓ y) (x ⊔ y) = dist x y 
goal: dist (x ⊓ y) (x ⊔ y) = dist x y

full_name: dist_mono_left_pi
code: lemma dist_mono_left_pi : MonotoneOn (dist · y) (Ici y) 
goal: MonotoneOn (fun x => dist x y) (Set.Ici y)

full_name: dist_mono_right_pi
code: lemma dist_mono_right_pi : MonotoneOn (dist x) (Ici x) 
goal: MonotoneOn (dist x) (Set.Ici x)

full_name: dist_anti_left_pi
code: lemma dist_anti_left_pi : AntitoneOn (dist · y) (Iic y) 
goal: AntitoneOn (fun x => dist x y) (Set.Iic y)

full_name: dist_anti_right_pi
code: lemma dist_anti_right_pi : AntitoneOn (dist x) (Iic x) 
goal: AntitoneOn (dist x) (Set.Iic x)

full_name: dist_le_dist_of_le_pi
code: lemma dist_le_dist_of_le_pi (ha : a₂ ≤ a₁) (h₁ : a₁ ≤ b₁) (hb : b₁ ≤ b₂) :
    dist a₁ b₁ ≤ dist a₂ b₂ 
goal: dist a₁ b₁ ≤ dist a₂ b₂

full_name: IsUpperSet.exists_subset_ball
code: theorem IsUpperSet.exists_subset_ball (hs : IsUpperSet s) (hx : x ∈ closure s) (hδ : 0 < δ) :
    ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s 
goal: ∃ y, Metric.closedBall y (δ / 4) ⊆ Metric.closedBall x δ ∧ Metric.closedBall y (δ / 4) ⊆ interior s

full_name: IsLowerSet.exists_subset_ball
code: theorem IsLowerSet.exists_subset_ball (hs : IsLowerSet s) (hx : x ∈ closure s) (hδ : 0 < δ) :
    ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s 
goal: ∃ y, Metric.closedBall y (δ / 4) ⊆ Metric.closedBall x δ ∧ Metric.closedBall y (δ / 4) ⊆ interior s

full_name: IsClosed.upperClosure_pi
code: protected lemma IsClosed.upperClosure_pi (hs : IsClosed s) (hs' : BddBelow s) :
    IsClosed (upperClosure s : Set (ι → ℝ)) 
goal: IsClosed ↑(upperClosure s)

full_name: IsClosed.lowerClosure_pi
code: protected lemma IsClosed.lowerClosure_pi (hs : IsClosed s) (hs' : BddAbove s) :
    IsClosed (lowerClosure s : Set (ι → ℝ)) 
goal: IsClosed ↑(lowerClosure s)

full_name: IsClopen.upperClosure_pi
code: protected lemma IsClopen.upperClosure_pi (hs : IsClopen s) (hs' : BddBelow s) :
    IsClopen (upperClosure s : Set (ι → ℝ)) 
goal: IsClopen ↑(upperClosure s)

full_name: IsClopen.lowerClosure_pi
code: protected lemma IsClopen.lowerClosure_pi (hs : IsClopen s) (hs' : BddAbove s) :
    IsClopen (lowerClosure s : Set (ι → ℝ)) 
goal: IsClopen ↑(lowerClosure s)

full_name: closure_upperClosure_comm_pi
code: lemma closure_upperClosure_comm_pi (hs : BddBelow s) :
    closure (upperClosure s : Set (ι → ℝ)) = upperClosure (closure s) 
goal: closure ↑(upperClosure s) = ↑(upperClosure (closure s))

full_name: closure_lowerClosure_comm_pi
code: lemma closure_lowerClosure_comm_pi (hs : BddAbove s) :
    closure (lowerClosure s : Set (ι → ℝ)) = lowerClosure (closure s) 
goal: closure ↑(lowerClosure s) = ↑(lowerClosure (closure s))