File: Mathlib/Topology/Bornology/BoundedOperation.lean

full_name: BoundedAdd.isBounded_add
code: /-- A typeclass saying that `(p : R × R) ↦ p.1 + p.2` maps any product of bounded sets to a bounded
set. This property follows from `LipschitzAdd`, and thus automatically holds, e.g., for seminormed
additive groups. -/
class BoundedAdd (R : Type*) [Bornology R] [Add R] : Prop where
  isBounded_add : ∀ {s t : Set R},
    Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s + t)
goal: Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s + t)

full_name: BoundedAdd
code: /-- A typeclass saying that `(p : R × R) ↦ p.1 + p.2` maps any product of bounded sets to a bounded
set. This property follows from `LipschitzAdd`, and thus automatically holds, e.g., for seminormed
additive groups. -/
class BoundedAdd (R : Type*) [Bornology R] [Add R] : Prop where
  isBounded_add : ∀ {s t : Set R},
    Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s + t)
goal: Prop

full_name: isBounded_add
code: lemma isBounded_add [Bornology R] [Add R] [BoundedAdd R] {s t : Set R}
    (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t) :
    Bornology.IsBounded (s + t) 
goal: Bornology.IsBounded (s + t)

full_name: add_bounded_of_bounded_of_bounded
code: lemma add_bounded_of_bounded_of_bounded {X : Type*} [PseudoMetricSpace R] [Add R] [BoundedAdd R]
    {f g : X → R} (f_bdd : ∃ C, ∀ x y, dist (f x) (f y) ≤ C)
    (g_bdd : ∃ C, ∀ x y, dist (g x) (g y) ≤ C) :
    ∃ C, ∀ x y, dist ((f + g) x) ((f + g) y) ≤ C 
goal: ∃ C, ∀ (x y : X), dist ((f + g) x) ((f + g) y) ≤ C

full_name: instBoundedAddOfLipschitzAdd
code: instance [PseudoMetricSpace R] [AddMonoid R] [LipschitzAdd R] : BoundedAdd R 
goal: BoundedAdd R

full_name: BoundedSub
code: /-- A typeclass saying that `(p : R × R) ↦ p.1 - p.2` maps any product of bounded sets to a bounded
set. This property automatically holds for seminormed additive groups, but it also holds, e.g.,
for `ℝ≥0`. -/
class BoundedSub (R : Type*) [Bornology R] [Sub R] : Prop where
  isBounded_sub : ∀ {s t : Set R},
    Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s - t)
goal: Prop

full_name: BoundedSub.isBounded_sub
code: /-- A typeclass saying that `(p : R × R) ↦ p.1 - p.2` maps any product of bounded sets to a bounded
set. This property automatically holds for seminormed additive groups, but it also holds, e.g.,
for `ℝ≥0`. -/
class BoundedSub (R : Type*) [Bornology R] [Sub R] : Prop where
  isBounded_sub : ∀ {s t : Set R},
    Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s - t)
goal: Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s - t)

full_name: isBounded_sub
code: lemma isBounded_sub [Bornology R] [Sub R] [BoundedSub R] {s t : Set R}
    (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t) :
    Bornology.IsBounded (s - t) 
goal: Bornology.IsBounded (s - t)

full_name: sub_bounded_of_bounded_of_bounded
code: lemma sub_bounded_of_bounded_of_bounded {X : Type*} [PseudoMetricSpace R] [Sub R] [BoundedSub R]
    {f g : X → R} (f_bdd : ∃ C, ∀ x y, dist (f x) (f y) ≤ C)
    (g_bdd : ∃ C, ∀ x y, dist (g x) (g y) ≤ C) :
    ∃ C, ∀ x y, dist ((f - g) x) ((f - g) y) ≤ C 
goal: ∃ C, ∀ (x y : X), dist ((f - g) x) ((f - g) y) ≤ C

full_name: boundedSub_of_lipschitzWith_sub
code: lemma boundedSub_of_lipschitzWith_sub [PseudoMetricSpace R] [Sub R] {K : NNReal}
    (lip : LipschitzWith K (fun (p : R × R) ↦ p.1 - p.2)) :
    BoundedSub R 
goal: BoundedSub R

full_name: BoundedMul.isBounded_mul
code: /-- A typeclass saying that `(p : R × R) ↦ p.1 * p.2` maps any product of bounded sets to a bounded
set. This property automatically holds for non-unital seminormed rings, but it also holds, e.g.,
for `ℝ≥0`. -/
class BoundedMul (R : Type*) [Bornology R] [Mul R] : Prop where
  isBounded_mul : ∀ {s t : Set R},
    Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s * t)
goal: Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s * t)

full_name: BoundedMul
code: /-- A typeclass saying that `(p : R × R) ↦ p.1 * p.2` maps any product of bounded sets to a bounded
set. This property automatically holds for non-unital seminormed rings, but it also holds, e.g.,
for `ℝ≥0`. -/
class BoundedMul (R : Type*) [Bornology R] [Mul R] : Prop where
  isBounded_mul : ∀ {s t : Set R},
    Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s * t)
goal: Prop

full_name: isBounded_mul
code: lemma isBounded_mul [Bornology R] [Mul R] [BoundedMul R] {s t : Set R}
    (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t) :
    Bornology.IsBounded (s * t) 
goal: Bornology.IsBounded (s * t)

full_name: isBounded_pow
code: lemma isBounded_pow {R : Type*} [Bornology R] [Monoid R] [BoundedMul R] {s : Set R}
    (s_bdd : Bornology.IsBounded s) (n : ℕ) :
    Bornology.IsBounded ((fun x ↦ x ^ n) '' s) 
goal: Bornology.IsBounded ((fun x => x ^ n) '' s)

full_name: mul_bounded_of_bounded_of_bounded
code: lemma mul_bounded_of_bounded_of_bounded {X : Type*} [PseudoMetricSpace R] [Mul R] [BoundedMul R]
    {f g : X → R} (f_bdd : ∃ C, ∀ x y, dist (f x) (f y) ≤ C)
    (g_bdd : ∃ C, ∀ x y, dist (g x) (g y) ≤ C) :
    ∃ C, ∀ x y, dist ((f * g) x) ((f * g) y) ≤ C 
goal: ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C

full_name: SeminormedAddCommGroup.lipschitzWith_sub
code: lemma SeminormedAddCommGroup.lipschitzWith_sub :
    LipschitzWith 2 (fun (p : R × R) ↦ p.1 - p.2) 
goal: LipschitzWith 2 fun p => p.1 - p.2

full_name: instBoundedSub
code: instance : BoundedSub R 
goal: BoundedSub R

full_name: instBoundedMul
code: instance : BoundedMul R 
goal: BoundedMul R

full_name: instBoundedSubNNReal
code: instance : BoundedSub ℝ≥0 
goal: BoundedSub ℝ≥0

full_name: instBoundedMulNNReal
code: instance : BoundedMul ℝ≥0 
goal: BoundedMul ℝ≥0