ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.Closeds.coe_finset_sup
@[simp, norm_cast]
theorem coe_finset_sup (f : ι → Closeds α) (s : Finset ι) :
    (↑(s.sup f) : Set α) = s.sup ((↑) ∘ f) 
Arguments: {ι : Type u_1} {α : Type u_2} [TopologicalSpace α] (f : ι → TopologicalSpace.Closeds α) (s : Finset ι)
Goal: ↑(s.sup f) = s.sup (SetLike.coe ∘ f)
-- TopologicalSpace.Closeds.coe_finset_inf
@[simp, norm_cast]
theorem coe_finset_inf (f : ι → Closeds α) (s : Finset ι) :
    (↑(s.inf f) : Set α) = s.inf ((↑) ∘ f) 
Arguments: {ι : Type u_1} {α : Type u_2} [TopologicalSpace α] (f : ι → TopologicalSpace.Closeds α) (s : Finset ι)
Goal: ↑(s.inf f) = s.inf (SetLike.coe ∘ f)
-- TopologicalSpace.Closeds.mem_sInf
@[simp]
theorem mem_sInf {S : Set (Closeds α)} {x : α} : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s 
Arguments: {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)} {x : α}
Goal: x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s
-- TopologicalSpace.Closeds.mem_iInf
@[simp]
theorem mem_iInf {ι} {x : α} {s : ι → Closeds α} : x ∈ iInf s ↔ ∀ i, x ∈ s i 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} {x : α} {s : ι → TopologicalSpace.Closeds α}
Goal: x ∈ iInf s ↔ ∀ (i : ι), x ∈ s i
-- TopologicalSpace.Closeds.coe_iInf
@[simp, norm_cast]
theorem coe_iInf {ι} (s : ι → Closeds α) : ((⨅ i, s i : Closeds α) : Set α) = ⋂ i, s i 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} (s : ι → TopologicalSpace.Closeds α)
Goal: ↑(⨅ i, s i) = ⋂ i, ↑(s i)
-- TopologicalSpace.Closeds.iInf_def
theorem iInf_def {ι} (s : ι → Closeds α) :
    ⨅ i, s i = ⟨⋂ i, s i, isClosed_iInter fun i => (s i).2⟩ 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} (s : ι → TopologicalSpace.Closeds α)
Goal: ⨅ i, s i = { carrier := ⋂ i, ↑(s i), closed' := ⋯ }
-- TopologicalSpace.Closeds.iInf_mk
@[simp]
theorem iInf_mk {ι} (s : ι → Set α) (h : ∀ i, IsClosed (s i)) :
    (⨅ i, ⟨s i, h i⟩ : Closeds α) = ⟨⋂ i, s i, isClosed_iInter h⟩ 
Arguments: {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} (s : ι → Set α) (h : ∀ (i : ι), IsClosed (s i))
Goal: ⨅ i, { carrier := s i, closed' := ⋯ } = { carrier := ⋂ i, s i, closed' := ⋯ }
-- TopologicalSpace.Closeds.coe_singleton
@[simps]
def singleton [T1Space α] (x : α) : Closeds α 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] (x : α)
Goal: ↑(TopologicalSpace.Closeds.singleton x) = {x}
-- TopologicalSpace.Closeds.mem_singleton
@[simp] lemma mem_singleton [T1Space α] {a b : α} : a ∈ singleton b ↔ a = b 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] {a b : α}
Goal: a ∈ TopologicalSpace.Closeds.singleton b ↔ a = b
-- TopologicalSpace.Closeds.compl_coe
@[simps]
def Closeds.compl (s : Closeds α) : Opens α 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α)
Goal: ↑s.compl = (↑s)ᶜ
-- TopologicalSpace.Opens.coe_compl
@[simps]
def Opens.compl (s : Opens α) : Closeds α 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Opens α)
Goal: ↑s.compl = (↑s)ᶜ
-- TopologicalSpace.Closeds.compl_compl
nonrec theorem Closeds.compl_compl (s : Closeds α) : s.compl.compl = s 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α)
Goal: s.compl.compl = s
-- TopologicalSpace.Opens.compl_compl
nonrec theorem Opens.compl_compl (s : Opens α) : s.compl.compl = s 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Opens α)
Goal: s.compl.compl = s
-- TopologicalSpace.Closeds.compl_bijective
theorem Closeds.compl_bijective : Function.Bijective (@Closeds.compl α _) 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: Function.Bijective TopologicalSpace.Closeds.compl
-- TopologicalSpace.Opens.compl_bijective
theorem Opens.compl_bijective : Function.Bijective (@Opens.compl α _) 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: Function.Bijective TopologicalSpace.Opens.compl
-- TopologicalSpace.Closeds.complOrderIso_apply
@[simps]
def Closeds.complOrderIso : Closeds α ≃o (Opens α)ᵒᵈ 
Arguments: (α : Type u_2) [TopologicalSpace α]
Goal: ∀ (a : TopologicalSpace.Closeds α),
  (TopologicalSpace.Closeds.complOrderIso α) a = (⇑OrderDual.toDual ∘ TopologicalSpace.Closeds.compl) a
-- TopologicalSpace.Closeds.complOrderIso_symm_apply
@[simps]
def Closeds.complOrderIso : Closeds α ≃o (Opens α)ᵒᵈ 
Arguments: (α : Type u_2) [TopologicalSpace α]
Goal: ∀ (a : (TopologicalSpace.Opens α)ᵒᵈ),
  (RelIso.symm (TopologicalSpace.Closeds.complOrderIso α)) a = (TopologicalSpace.Opens.compl ∘ ⇑OrderDual.ofDual) a
-- TopologicalSpace.Opens.complOrderIso_symm_apply
@[simps]
def Opens.complOrderIso : Opens α ≃o (Closeds α)ᵒᵈ 
Arguments: (α : Type u_2) [TopologicalSpace α]
Goal: ∀ (a : (TopologicalSpace.Closeds α)ᵒᵈ),
  (RelIso.symm (TopologicalSpace.Opens.complOrderIso α)) a = (TopologicalSpace.Closeds.compl ∘ ⇑OrderDual.ofDual) a
-- TopologicalSpace.Opens.complOrderIso_apply
@[simps]
def Opens.complOrderIso : Opens α ≃o (Closeds α)ᵒᵈ 
Arguments: (α : Type u_2) [TopologicalSpace α]
Goal: ∀ (a : TopologicalSpace.Opens α),
  (TopologicalSpace.Opens.complOrderIso α) a = (⇑OrderDual.toDual ∘ TopologicalSpace.Opens.compl) a
-- TopologicalSpace.Closeds.coe_eq_singleton_of_isAtom
lemma Closeds.coe_eq_singleton_of_isAtom [T0Space α] {s : Closeds α} (hs : IsAtom s) :
    ∃ a, (s : Set α) = {a} 
Arguments: {α : Type u_2} [TopologicalSpace α] [T0Space α] {s : TopologicalSpace.Closeds α} (hs : IsAtom s)
Goal: ∃ a, ↑s = {a}
-- TopologicalSpace.Closeds.isAtom_coe
@[simp, norm_cast] lemma Closeds.isAtom_coe [T1Space α] {s : Closeds α} :
    IsAtom (s : Set α) ↔ IsAtom s 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] {s : TopologicalSpace.Closeds α}
Goal: IsAtom ↑s ↔ IsAtom s
-- TopologicalSpace.Closeds.isAtom_iff
theorem Closeds.isAtom_iff [T1Space α] {s : Closeds α} :
    IsAtom s ↔ ∃ x, s = Closeds.singleton x 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] {s : TopologicalSpace.Closeds α}
Goal: IsAtom s ↔ ∃ x, s = TopologicalSpace.Closeds.singleton x
-- TopologicalSpace.Opens.isCoatom_iff
theorem Opens.isCoatom_iff [T1Space α] {s : Opens α} :
    IsCoatom s ↔ ∃ x, s = (Closeds.singleton x).compl 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] {s : TopologicalSpace.Opens α}
Goal: IsCoatom s ↔ ∃ x, s = (TopologicalSpace.Closeds.singleton x).compl
-- TopologicalSpace.Clopens.isClopen'
structure Clopens (α : Type*) [TopologicalSpace α] where
    carrier : Set α
  isClopen' : IsClopen carrier
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.Clopens α)
Goal: IsClopen self.carrier
-- TopologicalSpace.Clopens.isClopen
theorem isClopen (s : Clopens α) : IsClopen (s : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α)
Goal: IsClopen ↑s
-- TopologicalSpace.Clopens.toOpens_coe
@[simps]
def toOpens (s : Clopens α) : Opens α 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α)
Goal: ↑s.toOpens = ↑s
-- TopologicalSpace.Clopens.ext
@[ext]
protected theorem ext {s t : Clopens α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_2} [TopologicalSpace α] {s t : TopologicalSpace.Clopens α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.Clopens.coe_mk
@[simp]
theorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : Set α) (h : IsClopen s)
Goal: ↑{ carrier := s, isClopen' := h } = s
-- TopologicalSpace.Clopens.mem_mk
@[simp] lemma mem_mk {s : Set α} {x h} : x ∈ mk s h ↔ x ∈ s 
Arguments: {α : Type u_2} [TopologicalSpace α] {s : Set α} {x : α} {h : IsClopen s}
Goal: x ∈ { carrier := s, isClopen' := h } ↔ x ∈ s
-- TopologicalSpace.Clopens.coe_sup
@[simp, norm_cast] lemma coe_sup (s t : Clopens α) : ↑(s ⊔ t) = (s ∪ t : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Clopens α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.Clopens.coe_inf
@[simp, norm_cast] lemma coe_inf (s t : Clopens α) : ↑(s ⊓ t) = (s ∩ t : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Clopens α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- TopologicalSpace.Clopens.coe_top
@[simp, norm_cast] lemma coe_top : (↑(⊤ : Clopens α) : Set α) = univ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.Clopens.coe_bot
@[simp, norm_cast] lemma coe_bot : (↑(⊥ : Clopens α) : Set α) = ∅ 
Arguments: {α : Type u_2} [TopologicalSpace α]
Goal: ↑⊥ = ∅
-- TopologicalSpace.Clopens.coe_sdiff
@[simp, norm_cast] lemma coe_sdiff (s t : Clopens α) : ↑(s \ t) = (s \ t : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Clopens α)
Goal: ↑(s \ t) = ↑s \ ↑t
-- TopologicalSpace.Clopens.coe_himp
@[simp, norm_cast] lemma coe_himp (s t : Clopens α) : ↑(s ⇨ t) = (s ⇨ t : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s t : TopologicalSpace.Clopens α)
Goal: ↑(s ⇨ t) = ↑s ⇨ ↑t
-- TopologicalSpace.Clopens.coe_compl
@[simp, norm_cast] lemma coe_compl (s : Clopens α) : (↑sᶜ : Set α) = (↑s)ᶜ 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α)
Goal: ↑sᶜ = (↑s)ᶜ
-- TopologicalSpace.Clopens.mem_prod
@[simp]
protected lemma mem_prod {s : Clopens α} {t : Clopens β} {x : α × β} :
    x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {s : TopologicalSpace.Clopens α} {t : TopologicalSpace.Clopens β} {x : α × β}
Goal: x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t
-- TopologicalSpace.IrreducibleCloseds.is_closed'
structure IrreducibleCloseds (α : Type*) [TopologicalSpace α] where
    carrier : Set α
  is_irreducible' : IsIrreducible carrier
  is_closed' : IsClosed carrier
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.IrreducibleCloseds α)
Goal: IsClosed self.carrier
-- TopologicalSpace.IrreducibleCloseds.is_irreducible'
structure IrreducibleCloseds (α : Type*) [TopologicalSpace α] where
    carrier : Set α
  is_irreducible' : IsIrreducible carrier
  is_closed' : IsClosed carrier
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.IrreducibleCloseds α)
Goal: IsIrreducible self.carrier
-- TopologicalSpace.IrreducibleCloseds.isIrreducible
theorem isIrreducible (s : IrreducibleCloseds α) : IsIrreducible (s : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.IrreducibleCloseds α)
Goal: IsIrreducible ↑s
-- TopologicalSpace.IrreducibleCloseds.isClosed
theorem isClosed (s : IrreducibleCloseds α) : IsClosed (s : Set α) 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.IrreducibleCloseds α)
Goal: IsClosed ↑s
-- TopologicalSpace.IrreducibleCloseds.ext
@[ext]
protected theorem ext {s t : IrreducibleCloseds α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_2} [TopologicalSpace α] {s t : TopologicalSpace.IrreducibleCloseds α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.IrreducibleCloseds.coe_mk
@[simp]
theorem coe_mk (s : Set α) (h : IsIrreducible s) (h' : IsClosed s) : (mk s h h' : Set α) = s 
Arguments: {α : Type u_2} [TopologicalSpace α] (s : Set α) (h : IsIrreducible s) (h' : IsClosed s)
Goal: ↑{ carrier := s, is_irreducible' := h, is_closed' := h' } = s
-- TopologicalSpace.IrreducibleCloseds.coe_singleton
@[simps]
def singleton [T1Space α] (x : α) : IrreducibleCloseds α 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] (x : α)
Goal: ↑(TopologicalSpace.IrreducibleCloseds.singleton x) = {x}
-- TopologicalSpace.IrreducibleCloseds.mem_singleton
@[simp] lemma mem_singleton [T1Space α] {a b : α} : a ∈ singleton b ↔ a = b 
Arguments: {α : Type u_2} [TopologicalSpace α] [T1Space α] {a b : α}
Goal: a ∈ TopologicalSpace.IrreducibleCloseds.singleton b ↔ a = b
-- TopologicalSpace.IrreducibleCloseds.equivSubtype_symm_apply
@[simps apply symm_apply]
def equivSubtype : IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x } 
Arguments: {α : Type u_2} [TopologicalSpace α] (a : { x // IsIrreducible x ∧ IsClosed x })
Goal: TopologicalSpace.IrreducibleCloseds.equivSubtype.symm a = { carrier := ↑a, is_irreducible' := ⋯, is_closed' := ⋯ }
-- TopologicalSpace.IrreducibleCloseds.equivSubtype_apply
@[simps apply symm_apply]
def equivSubtype : IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x } 
Arguments: {α : Type u_2} [TopologicalSpace α] (a : TopologicalSpace.IrreducibleCloseds α)
Goal: TopologicalSpace.IrreducibleCloseds.equivSubtype a = ⟨a.carrier, ⋯⟩
-- TopologicalSpace.IrreducibleCloseds.equivSubtype'_apply
@[simps apply symm_apply]
def equivSubtype' : IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x } 
Arguments: {α : Type u_2} [TopologicalSpace α] (a : TopologicalSpace.IrreducibleCloseds α)
Goal: TopologicalSpace.IrreducibleCloseds.equivSubtype' a = ⟨a.carrier, ⋯⟩
-- TopologicalSpace.IrreducibleCloseds.equivSubtype'_symm_apply
@[simps apply symm_apply]
def equivSubtype' : IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x } 
Arguments: {α : Type u_2} [TopologicalSpace α] (a : { x // IsClosed x ∧ IsIrreducible x })
Goal: TopologicalSpace.IrreducibleCloseds.equivSubtype'.symm a = { carrier := ↑a, is_irreducible' := ⋯, is_closed' := ⋯ }
-- BoundedAdd.isBounded_add
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)
Arguments: {R : Type u_1} [Bornology R] [Add R] [self : BoundedAdd R] {s t : Set R}
Goal: Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s + t)
-- isBounded_add
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) 
Arguments: {R : Type u_1} [Bornology R] [Add R] [BoundedAdd R] {s t : Set R} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s + t)
-- add_bounded_of_bounded_of_bounded
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 
Arguments: {R : Type u_1} {X : Type u_2} [PseudoMetricSpace R] [Add R] [BoundedAdd R] {f g : X → R} (f_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C) (g_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C)
Goal: ∃ C, ∀ (x y : X), dist ((f + g) x) ((f + g) y) ≤ C
-- BoundedSub.isBounded_sub
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)
Arguments: {R : Type u_1} [Bornology R] [Sub R] [self : BoundedSub R] {s t : Set R}
Goal: Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s - t)
-- isBounded_sub
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) 
Arguments: {R : Type u_1} [Bornology R] [Sub R] [BoundedSub R] {s t : Set R} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s - t)
-- sub_bounded_of_bounded_of_bounded
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 
Arguments: {R : Type u_1} {X : Type u_2} [PseudoMetricSpace R] [Sub R] [BoundedSub R] {f g : X → R} (f_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C) (g_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C)
Goal: ∃ C, ∀ (x y : X), dist ((f - g) x) ((f - g) y) ≤ C
-- boundedSub_of_lipschitzWith_sub
lemma boundedSub_of_lipschitzWith_sub [PseudoMetricSpace R] [Sub R] {K : NNReal}
    (lip : LipschitzWith K (fun (p : R × R) ↦ p.1 - p.2)) :
    BoundedSub R 
Arguments: {R : Type u_1} [PseudoMetricSpace R] [Sub R] {K : ℝ≥0} (lip : LipschitzWith K fun p => p.1 - p.2)
Goal: BoundedSub R
-- BoundedMul.isBounded_mul
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)
Arguments: {R : Type u_1} [Bornology R] [Mul R] [self : BoundedMul R] {s t : Set R}
Goal: Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s * t)
-- isBounded_mul
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) 
Arguments: {R : Type u_1} [Bornology R] [Mul R] [BoundedMul R] {s t : Set R} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s * t)
-- isBounded_pow
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) 
Arguments: {R : Type u_2} [Bornology R] [Monoid R] [BoundedMul R] {s : Set R} (s_bdd : Bornology.IsBounded s) (n : ℕ)
Goal: Bornology.IsBounded ((fun x => x ^ n) '' s)
-- mul_bounded_of_bounded_of_bounded
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 
Arguments: {R : Type u_1} {X : Type u_2} [PseudoMetricSpace R] [Mul R] [BoundedMul R] {f g : X → R} (f_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C) (g_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C)
Goal: ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C
-- SeminormedAddCommGroup.lipschitzWith_sub
lemma SeminormedAddCommGroup.lipschitzWith_sub :
    LipschitzWith 2 (fun (p : R × R) ↦ p.1 - p.2) 
Arguments: {R : Type u_1} [SeminormedAddCommGroup R]
Goal: LipschitzWith 2 fun p => p.1 - p.2
-- TopologicalSpace.Compacts.isCompact'
structure Compacts (α : Type*) [TopologicalSpace α] where
    carrier : Set α
  isCompact' : IsCompact carrier
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.Compacts α)
Goal: IsCompact self.carrier
-- TopologicalSpace.Compacts.isCompact
protected theorem isCompact (s : Compacts α) : IsCompact (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α)
Goal: IsCompact ↑s
-- TopologicalSpace.Compacts.ext
@[ext]
protected theorem ext {s t : Compacts α} (h : (s : Set α) = t) : s = t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : TopologicalSpace.Compacts α} (h : ↑s = ↑t)
Goal: s = t
-- TopologicalSpace.Compacts.coe_mk
@[simp]
theorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) (h : IsCompact s)
Goal: ↑{ carrier := s, isCompact' := h } = s
-- TopologicalSpace.Compacts.carrier_eq_coe
@[simp]
theorem carrier_eq_coe (s : Compacts α) : s.carrier = s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α)
Goal: s.carrier = ↑s
-- TopologicalSpace.Compacts.coe_sup
@[simp]
theorem coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t 
Arguments: {α : Type u_1} [TopologicalSpace α] (s t : TopologicalSpace.Compacts α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- TopologicalSpace.Compacts.coe_inf
@[simp]
theorem coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] (s t : TopologicalSpace.Compacts α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- TopologicalSpace.Compacts.coe_top
@[simp]
theorem coe_top [CompactSpace α] : (↑(⊤ : Compacts α) : Set α) = univ 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α]
Goal: ↑⊤ = Set.univ
-- TopologicalSpace.Compacts.coe_bot
@[simp]
theorem coe_bot : (↑(⊥ : Compacts α) : Set α) = ∅ 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: ↑⊥ = ∅
-- TopologicalSpace.Compacts.coe_finset_sup
@[simp]
theorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :
    (↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_4} {s : Finset ι} {f : ι → TopologicalSpace.Compacts α}
Goal: ↑(s.sup f) = s.sup fun i => ↑(f i)
-- TopologicalSpace.Compacts.coe_map
@[simp, norm_cast]
theorem coe_map {f : α → β} (hf : Continuous f) (s : Compacts α) : (s.map f hf : Set β) = f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Continuous f) (s : TopologicalSpace.Compacts α)
Goal: ↑(TopologicalSpace.Compacts.map f hf s) = f '' ↑s
-- TopologicalSpace.Compacts.map_id
@[simp]
theorem map_id (K : Compacts α) : K.map id continuous_id = K 
Arguments: {α : Type u_1} [TopologicalSpace α] (K : TopologicalSpace.Compacts α)
Goal: TopologicalSpace.Compacts.map id ⋯ K = K
-- TopologicalSpace.Compacts.map_comp
theorem map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (K : Compacts α) :
    K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (K : TopologicalSpace.Compacts α)
Goal: TopologicalSpace.Compacts.map (f ∘ g) ⋯ K = TopologicalSpace.Compacts.map f hf (TopologicalSpace.Compacts.map g hg K)
-- TopologicalSpace.Compacts.equiv_symm_apply
@[simps]
protected def equiv (f : α ≃ₜ β) : Compacts α ≃ Compacts β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts β)
Goal: (TopologicalSpace.Compacts.equiv f).symm K = TopologicalSpace.Compacts.map ⇑f.symm ⋯ K
-- TopologicalSpace.Compacts.equiv_apply
@[simps]
protected def equiv (f : α ≃ₜ β) : Compacts α ≃ Compacts β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts α)
Goal: (TopologicalSpace.Compacts.equiv f) K = TopologicalSpace.Compacts.map ⇑f ⋯ K
-- TopologicalSpace.Compacts.equiv_refl
@[simp]
theorem equiv_refl : Compacts.equiv (Homeomorph.refl α) = Equiv.refl _ 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: TopologicalSpace.Compacts.equiv (Homeomorph.refl α) = Equiv.refl (TopologicalSpace.Compacts α)
-- TopologicalSpace.Compacts.equiv_trans
@[simp]
theorem equiv_trans (f : α ≃ₜ β) (g : β ≃ₜ γ) :
    Compacts.equiv (f.trans g) = (Compacts.equiv f).trans (Compacts.equiv g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : α ≃ₜ β) (g : β ≃ₜ γ)
Goal: TopologicalSpace.Compacts.equiv (f.trans g) =
  (TopologicalSpace.Compacts.equiv f).trans (TopologicalSpace.Compacts.equiv g)
-- TopologicalSpace.Compacts.equiv_symm
@[simp]
theorem equiv_symm (f : α ≃ₜ β) : Compacts.equiv f.symm = (Compacts.equiv f).symm 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β)
Goal: TopologicalSpace.Compacts.equiv f.symm = (TopologicalSpace.Compacts.equiv f).symm
-- TopologicalSpace.Compacts.coe_equiv_apply_eq_preimage
theorem coe_equiv_apply_eq_preimage (f : α ≃ₜ β) (K : Compacts α) :
    (Compacts.equiv f K : Set β) = f.symm ⁻¹' (K : Set α) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts α)
Goal: ↑((TopologicalSpace.Compacts.equiv f) K) = ⇑f.symm ⁻¹' ↑K
-- TopologicalSpace.Compacts.coe_prod
@[simp]
theorem coe_prod (K : Compacts α) (L : Compacts β) :
    (K.prod L : Set (α × β)) = (K : Set α) ×ˢ (L : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (K : TopologicalSpace.Compacts α) (L : TopologicalSpace.Compacts β)
Goal: ↑(K.prod L) = ↑K ×ˢ ↑L
-- TopologicalSpace.NonemptyCompacts.nonempty'
structure NonemptyCompacts (α : Type*) [TopologicalSpace α] extends Compacts α where
  nonempty' : carrier.Nonempty
Arguments: {α : Type u_4} [TopologicalSpace α] (self : TopologicalSpace.NonemptyCompacts α)
Goal: self.carrier.Nonempty
-- TopologicalSpace.NonemptyCompacts.isCompact
protected theorem isCompact (s : NonemptyCompacts α) : IsCompact (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.NonemptyCompacts α)
Goal: IsCompact ↑s
-- TopologicalSpace.NonemptyCompacts.nonempty
protected theorem nonempty (s : NonemptyCompacts α) : (s : Set α).Nonempty 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.NonemptyCompacts α)
Goal: (↑s).Nonempty