ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.IsTopologicalBasis.exists_closure_subset
theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set X)}
    (hB : IsTopologicalBasis B) {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
    ∃ t ∈ B, x ∈ t ∧ closure t ⊆ s 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {B : Set (Set X)} (hB : TopologicalSpace.IsTopologicalBasis B) {x : X} {s : Set X} (h : s ∈ 𝓝 x)
Goal: ∃ t ∈ B, x ∈ t ∧ closure t ⊆ s
-- Inducing.regularSpace
protected theorem Inducing.regularSpace [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) :
    RegularSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [RegularSpace X] [TopologicalSpace Y] {f : Y → X} (hf : Inducing f)
Goal: RegularSpace Y
-- regularSpace_induced
theorem regularSpace_induced (f : Y → X) : @RegularSpace Y (induced f ‹_›) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [RegularSpace X] (f : Y → X)
Goal: RegularSpace Y
-- regularSpace_sInf
theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
    @RegularSpace X (sInf T) 
Arguments: {X : Type u_3} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, RegularSpace X)
Goal: RegularSpace X
-- regularSpace_iInf
theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
    @RegularSpace X (iInf t) 
Arguments: {ι : Sort u_3} {X : Type u_4} {t : ι → TopologicalSpace X} (h : ∀ (i : ι), RegularSpace X)
Goal: RegularSpace X
-- RegularSpace.inf
theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
    (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) 
Arguments: {X : Type u_3} {t₁ t₂ : TopologicalSpace X} (h₁ : RegularSpace X) (h₂ : RegularSpace X)
Goal: RegularSpace X
-- SeparatedNhds.of_isCompact_isClosed
lemma SeparatedNhds.of_isCompact_isClosed {s t : Set X}
    (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) : SeparatedNhds s t 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X] {s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t)
Goal: SeparatedNhds s t
-- IsClosed.HasSeparatingCover
lemma IsClosed.HasSeparatingCover {s t : Set X} [r : RegularSpace X] [LindelofSpace X]
    (s_cl : IsClosed s) (t_cl : IsClosed t) (st_dis : Disjoint s t) : HasSeparatingCover s t 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} [r : RegularSpace X] [LindelofSpace X] (s_cl : IsClosed s) (t_cl : IsClosed t) (st_dis : Disjoint s t)
Goal: HasSeparatingCover s t
-- exists_compact_closed_between
theorem exists_compact_closed_between [LocallyCompactSpace X] [RegularSpace X]
    {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U) :
    ∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] [RegularSpace X] {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U)
Goal: ∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U
-- exists_open_between_and_isCompact_closure
theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [RegularSpace X]
    {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
    ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V) 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] [RegularSpace X] {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
Goal: ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V)
-- locally_compact_of_compact
@[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace (since := "2023-09-03")]
theorem locally_compact_of_compact [T2Space X] [CompactSpace X] :
    LocallyCompactSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X]
Goal: LocallyCompactSpace X
-- T25Space.t2_5
class T25Space (X : Type u) [TopologicalSpace X] : Prop where
    t2_5 : ∀ ⦃x y : X⦄, x ≠ y → Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure)
Arguments: {X : Type u} [TopologicalSpace X] [self : T25Space X] ⦃x y : X⦄
Goal: x ≠ y → Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure)
-- disjoint_lift'_closure_nhds
@[simp]
theorem disjoint_lift'_closure_nhds [T25Space X] {x y : X} :
    Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y 
Arguments: {X : Type u_1} [TopologicalSpace X] [T25Space X] {x y : X}
Goal: Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y
-- exists_nhds_disjoint_closure
theorem exists_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
    ∃ s ∈ 𝓝 x, ∃ t ∈ 𝓝 y, Disjoint (closure s) (closure t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T25Space X] {x y : X} (h : x ≠ y)
Goal: ∃ s ∈ 𝓝 x, ∃ t ∈ 𝓝 y, Disjoint (closure s) (closure t)
-- exists_open_nhds_disjoint_closure
theorem exists_open_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
    ∃ u : Set X,
      x ∈ u ∧ IsOpen u ∧ ∃ v : Set X, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T25Space X] {x y : X} (h : x ≠ y)
Goal: ∃ u, x ∈ u ∧ IsOpen u ∧ ∃ v, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v)
-- T25Space.of_injective_continuous
theorem T25Space.of_injective_continuous [TopologicalSpace Y] [T25Space Y] {f : X → Y}
    (hinj : Injective f) (hcont : Continuous f) : T25Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T25Space Y] {f : X → Y} (hinj : Function.Injective f) (hcont : Continuous f)
Goal: T25Space X
-- Embedding.t25Space
theorem Embedding.t25Space [TopologicalSpace Y] [T25Space Y] {f : X → Y} (hf : Embedding f) :
    T25Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T25Space Y] {f : X → Y} (hf : Embedding f)
Goal: T25Space X
-- RegularSpace.t3Space_iff_t0Space
theorem RegularSpace.t3Space_iff_t0Space [RegularSpace X] : T3Space X ↔ T0Space X 
Arguments: {X : Type u_1} [TopologicalSpace X] [RegularSpace X]
Goal: T3Space X ↔ T0Space X
-- Embedding.t3Space
protected theorem Embedding.t3Space [TopologicalSpace Y] [T3Space Y] {f : X → Y}
    (hf : Embedding f) : T3Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {f : X → Y} (hf : Embedding f)
Goal: T3Space X
-- disjoint_nested_nhds
theorem disjoint_nested_nhds [T3Space X] {x y : X} (h : x ≠ y) :
    ∃ U₁ ∈ 𝓝 x, ∃ V₁ ∈ 𝓝 x, ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y,
      IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] [T3Space X] {x y : X} (h : x ≠ y)
Goal: ∃ U₁ ∈ 𝓝 x,
  ∃ V₁ ∈ 𝓝 x,
    ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y, IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂
-- NormalSpace.normal
class NormalSpace (X : Type u) [TopologicalSpace X] : Prop where
    normal : ∀ s t : Set X, IsClosed s → IsClosed t → Disjoint s t → SeparatedNhds s t
Arguments: {X : Type u} [TopologicalSpace X] [self : NormalSpace X] (s t : Set X)
Goal: IsClosed s → IsClosed t → Disjoint s t → SeparatedNhds s t
-- normal_separation
theorem normal_separation [NormalSpace X] {s t : Set X} (H1 : IsClosed s) (H2 : IsClosed t)
    (H3 : Disjoint s t) : SeparatedNhds s t 
Arguments: {X : Type u_1} [TopologicalSpace X] [NormalSpace X] {s t : Set X} (H1 : IsClosed s) (H2 : IsClosed t) (H3 : Disjoint s t)
Goal: SeparatedNhds s t
-- disjoint_nhdsSet_nhdsSet
theorem disjoint_nhdsSet_nhdsSet [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsClosed t)
    (hd : Disjoint s t) : Disjoint (𝓝ˢ s) (𝓝ˢ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: Disjoint (𝓝ˢ s) (𝓝ˢ t)
-- normal_exists_closure_subset
theorem normal_exists_closure_subset [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsOpen t)
    (hst : s ⊆ t) : ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t 
Arguments: {X : Type u_1} [TopologicalSpace X] [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsOpen t) (hst : s ⊆ t)
Goal: ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t
-- ClosedEmbedding.normalSpace
protected theorem ClosedEmbedding.normalSpace [TopologicalSpace Y] [NormalSpace Y] {f : X → Y}
    (hf : ClosedEmbedding f) : NormalSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: NormalSpace X
-- T4Space.of_compactSpace_t2Space
@[deprecated inferInstance (since := "2024-01-28")]
theorem T4Space.of_compactSpace_t2Space [CompactSpace X] [T2Space X] :
    T4Space X 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] [T2Space X]
Goal: T4Space X
-- ClosedEmbedding.t4Space
protected theorem ClosedEmbedding.t4Space [TopologicalSpace Y] [T4Space Y] {f : X → Y}
    (hf : ClosedEmbedding f) : T4Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T4Space Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: T4Space X
-- CompletelyNormalSpace.completely_normal
class CompletelyNormalSpace (X : Type u) [TopologicalSpace X] : Prop where
    completely_normal :
    ∀ ⦃s t : Set X⦄, Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t)
Arguments: {X : Type u} [TopologicalSpace X] [self : CompletelyNormalSpace X] ⦃s t : Set X⦄
Goal: Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t)
-- Embedding.completelyNormalSpace
theorem Embedding.completelyNormalSpace [TopologicalSpace Y] [CompletelyNormalSpace Y] {e : X → Y}
    (he : Embedding e) : CompletelyNormalSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompletelyNormalSpace Y] {e : X → Y} (he : Embedding e)
Goal: CompletelyNormalSpace X
-- Embedding.t5Space
theorem Embedding.t5Space [TopologicalSpace Y] [T5Space Y] {e : X → Y} (he : Embedding e) :
    T5Space X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T5Space Y] {e : X → Y} (he : Embedding e)
Goal: T5Space X
-- PerfectlyNormalSpace.closed_gdelta
class PerfectlyNormalSpace (X : Type u) [TopologicalSpace X] extends NormalSpace X : Prop where
    closed_gdelta : ∀ ⦃h : Set X⦄, IsClosed h → IsGδ h
Arguments: {X : Type u} [TopologicalSpace X] [self : PerfectlyNormalSpace X] ⦃h : Set X⦄
Goal: IsClosed h → IsGδ h
-- Disjoint.hasSeparatingCover_closed_gdelta_right
theorem Disjoint.hasSeparatingCover_closed_gdelta_right {s t : Set X} [NormalSpace X]
    (st_dis : Disjoint s t) (t_cl : IsClosed t) (t_gd : IsGδ t) : HasSeparatingCover s t 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} [NormalSpace X] (st_dis : Disjoint s t) (t_cl : IsClosed t) (t_gd : IsGδ t)
Goal: HasSeparatingCover s t
-- connectedComponent_eq_iInter_isClopen
theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x : X) :
    connectedComponent x = ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X] (x : X)
Goal: connectedComponent x = ⋂ s, ↑s
-- totallySeparatedSpace_of_t1_of_basis_clopen
theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space X]
    (h : IsTopologicalBasis { s : Set X | IsClopen s }) : TotallySeparatedSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] [T1Space X] (h : TopologicalSpace.IsTopologicalBasis {s | IsClopen s})
Goal: TotallySeparatedSpace X
-- compact_t2_tot_disc_iff_tot_sep
theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace X ↔ TotallySeparatedSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X]
Goal: TotallyDisconnectedSpace X ↔ TotallySeparatedSpace X
-- nhds_basis_clopen
theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X] [TotallyDisconnectedSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => x ∈ s ∧ IsClopen s) id
-- isTopologicalBasis_isClopen
theorem isTopologicalBasis_isClopen : IsTopologicalBasis { s : Set X | IsClopen s } 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X] [TotallyDisconnectedSpace X]
Goal: TopologicalSpace.IsTopologicalBasis {s | IsClopen s}
-- compact_exists_isClopen_in_isOpen
theorem compact_exists_isClopen_in_isOpen {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U) :
    ∃ V : Set X, IsClopen V ∧ x ∈ V ∧ V ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [CompactSpace X] [TotallyDisconnectedSpace X] {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U)
Goal: ∃ V, IsClopen V ∧ x ∈ V ∧ V ⊆ U
-- loc_compact_Haus_tot_disc_of_zero_dim
theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
    IsTopologicalBasis { s : Set H | IsClopen s } 
Arguments: {H : Type u_3} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H] [TotallyDisconnectedSpace H]
Goal: TopologicalSpace.IsTopologicalBasis {s | IsClopen s}
-- loc_compact_t2_tot_disc_iff_tot_sep
theorem loc_compact_t2_tot_disc_iff_tot_sep :
    TotallyDisconnectedSpace H ↔ TotallySeparatedSpace H 
Arguments: {H : Type u_3} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H]
Goal: TotallyDisconnectedSpace H ↔ TotallySeparatedSpace H
-- ContinuousConstSMul.continuous_const_smul
class ContinuousConstSMul (Γ : Type*) (T : Type*) [TopologicalSpace T] [SMul Γ T] : Prop where
    continuous_const_smul : ∀ γ : Γ, Continuous fun x : T => γ • x
Arguments: {Γ : Type u_1} {T : Type u_2} [TopologicalSpace T] [SMul Γ T] [self : ContinuousConstSMul Γ T] (γ : Γ)
Goal: Continuous fun x => γ • x
-- ContinuousConstVAdd.continuous_const_vadd
class ContinuousConstVAdd (Γ : Type*) (T : Type*) [TopologicalSpace T] [VAdd Γ T] : Prop where
    continuous_const_vadd : ∀ γ : Γ, Continuous fun x : T => γ +ᵥ x
Arguments: {Γ : Type u_1} {T : Type u_2} [TopologicalSpace T] [VAdd Γ T] [self : ContinuousConstVAdd Γ T] (γ : Γ)
Goal: Continuous fun x => γ +ᵥ x
-- Filter.Tendsto.const_smul
@[to_additive]
theorem Filter.Tendsto.const_smul {f : β → α} {l : Filter β} {a : α} (hf : Tendsto f l (𝓝 a))
    (c : M) : Tendsto (fun x => c • f x) l (𝓝 (c • a)) 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] {f : β → α} {l : Filter β} {a : α} (hf : Filter.Tendsto f l (𝓝 a)) (c : M)
Goal: Filter.Tendsto (fun x => c • f x) l (𝓝 (c • a))
-- Filter.Tendsto.const_vadd
@[to_additive]
theorem Filter.Tendsto.const_smul {f : β → α} {l : Filter β} {a : α} (hf : Tendsto f l (𝓝 a))
    (c : M) : Tendsto (fun x => c • f x) l (𝓝 (c • a)) 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] {f : β → α} {l : Filter β} {a : α} (hf : Filter.Tendsto f l (𝓝 a)) (c : M)
Goal: Filter.Tendsto (fun x => c +ᵥ f x) l (𝓝 (c +ᵥ a))
-- ContinuousWithinAt.const_smul
@[to_additive]
nonrec theorem ContinuousWithinAt.const_smul (hg : ContinuousWithinAt g s b) (c : M) :
    ContinuousWithinAt (fun x => c • g x) s b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] [TopologicalSpace β] {g : β → α} {b : β} {s : Set β} (hg : ContinuousWithinAt g s b) (c : M)
Goal: ContinuousWithinAt (fun x => c • g x) s b
-- ContinuousWithinAt.const_vadd
@[to_additive]
nonrec theorem ContinuousWithinAt.const_smul (hg : ContinuousWithinAt g s b) (c : M) :
    ContinuousWithinAt (fun x => c • g x) s b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] [TopologicalSpace β] {g : β → α} {b : β} {s : Set β} (hg : ContinuousWithinAt g s b) (c : M)
Goal: ContinuousWithinAt (fun x => c +ᵥ g x) s b
-- ContinuousAt.const_vadd
@[to_additive (attr := fun_prop)]
nonrec theorem ContinuousAt.const_smul (hg : ContinuousAt g b) (c : M) :
    ContinuousAt (fun x => c • g x) b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] [TopologicalSpace β] {g : β → α} {b : β} (hg : ContinuousAt g b) (c : M)
Goal: ContinuousAt (fun x => c +ᵥ g x) b
-- ContinuousAt.const_smul
@[to_additive (attr := fun_prop)]
nonrec theorem ContinuousAt.const_smul (hg : ContinuousAt g b) (c : M) :
    ContinuousAt (fun x => c • g x) b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] [TopologicalSpace β] {g : β → α} {b : β} (hg : ContinuousAt g b) (c : M)
Goal: ContinuousAt (fun x => c • g x) b
-- ContinuousOn.const_smul
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.const_smul (hg : ContinuousOn g s) (c : M) :
    ContinuousOn (fun x => c • g x) s 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] [TopologicalSpace β] {g : β → α} {s : Set β} (hg : ContinuousOn g s) (c : M)
Goal: ContinuousOn (fun x => c • g x) s
-- ContinuousOn.const_vadd
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.const_smul (hg : ContinuousOn g s) (c : M) :
    ContinuousOn (fun x => c • g x) s 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] [TopologicalSpace β] {g : β → α} {s : Set β} (hg : ContinuousOn g s) (c : M)
Goal: ContinuousOn (fun x => c +ᵥ g x) s
-- Continuous.const_vadd
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.const_smul (hg : Continuous g) (c : M) : Continuous fun x => c • g x 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] [TopologicalSpace β] {g : β → α} (hg : Continuous g) (c : M)
Goal: Continuous fun x => c +ᵥ g x
-- Continuous.const_smul
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.const_smul (hg : Continuous g) (c : M) : Continuous fun x => c • g x 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] [TopologicalSpace β] {g : β → α} (hg : Continuous g) (c : M)
Goal: Continuous fun x => c • g x
-- IsCompact.vadd
@[to_additive]
theorem IsCompact.smul {α β} [SMul α β] [TopologicalSpace β] [ContinuousConstSMul α β] (a : α)
    {s : Set β} (hs : IsCompact s) : IsCompact (a • s) 
Arguments: {α : Type u_4} {β : Type u_5} [VAdd α β] [TopologicalSpace β] [ContinuousConstVAdd α β] (a : α) {s : Set β} (hs : IsCompact s)
Goal: IsCompact (a +ᵥ s)
-- IsCompact.smul
@[to_additive]
theorem IsCompact.smul {α β} [SMul α β] [TopologicalSpace β] [ContinuousConstSMul α β] (a : α)
    {s : Set β} (hs : IsCompact s) : IsCompact (a • s) 
Arguments: {α : Type u_4} {β : Type u_5} [SMul α β] [TopologicalSpace β] [ContinuousConstSMul α β] (a : α) {s : Set β} (hs : IsCompact s)
Goal: IsCompact (a • s)
-- Specializes.const_vadd
@[to_additive]
theorem Specializes.const_smul {x y : α} (h : x ⤳ y) (c : M) : (c • x) ⤳ (c • y) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] {x y : α} (h : x ⤳ y) (c : M)
Goal: (c +ᵥ x) ⤳ (c +ᵥ y)
-- Specializes.const_smul
@[to_additive]
theorem Specializes.const_smul {x y : α} (h : x ⤳ y) (c : M) : (c • x) ⤳ (c • y) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] {x y : α} (h : x ⤳ y) (c : M)
Goal: (c • x) ⤳ (c • y)
-- Inseparable.const_vadd
@[to_additive]
theorem Inseparable.const_smul {x y : α} (h : Inseparable x y) (c : M) :
    Inseparable (c • x) (c • y) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] {x y : α} (h : Inseparable x y) (c : M)
Goal: Inseparable (c +ᵥ x) (c +ᵥ y)
-- Inseparable.const_smul
@[to_additive]
theorem Inseparable.const_smul {x y : α} (h : Inseparable x y) (c : M) :
    Inseparable (c • x) (c • y) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] {x y : α} (h : Inseparable x y) (c : M)
Goal: Inseparable (c • x) (c • y)
-- smul_closure_subset
@[to_additive]
theorem smul_closure_subset (c : M) (s : Set α) : c • closure s ⊆ closure (c • s) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [Monoid M] [MulAction M α] [ContinuousConstSMul M α] (c : M) (s : Set α)
Goal: c • closure s ⊆ closure (c • s)
-- vadd_closure_subset
@[to_additive]
theorem smul_closure_subset (c : M) (s : Set α) : c • closure s ⊆ closure (c • s) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [AddMonoid M] [AddAction M α] [ContinuousConstVAdd M α] (c : M) (s : Set α)
Goal: c +ᵥ closure s ⊆ closure (c +ᵥ s)
-- smul_closure_orbit_subset
@[to_additive]
theorem smul_closure_orbit_subset (c : M) (x : α) :
    c • closure (MulAction.orbit M x) ⊆ closure (MulAction.orbit M x) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [Monoid M] [MulAction M α] [ContinuousConstSMul M α] (c : M) (x : α)
Goal: c • closure (MulAction.orbit M x) ⊆ closure (MulAction.orbit M x)
-- vadd_closure_orbit_subset
@[to_additive]
theorem smul_closure_orbit_subset (c : M) (x : α) :
    c • closure (MulAction.orbit M x) ⊆ closure (MulAction.orbit M x) 
Arguments: {M : Type u_1} {α : Type u_2} [TopologicalSpace α] [AddMonoid M] [AddAction M α] [ContinuousConstVAdd M α] (c : M) (x : α)
Goal: c +ᵥ closure (AddAction.orbit M x) ⊆ closure (AddAction.orbit M x)
-- isClosed_setOf_map_smul
theorem isClosed_setOf_map_smul {N : Type*} [Monoid N] (α β) [MulAction M α] [MulAction N β]
    [TopologicalSpace β] [T2Space β] [ContinuousConstSMul N β] (σ : M → N) :
    IsClosed { f : α → β | ∀ c x, f (c • x) = σ c • f x } 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_4} [Monoid N] (α : Type u_5) (β : Type u_6) [MulAction M α] [MulAction N β] [TopologicalSpace β] [T2Space β] [ContinuousConstSMul N β] (σ : M → N)
Goal: IsClosed {f | ∀ (c : M) (x : α), f (c • x) = σ c • f x}
-- tendsto_const_vadd_iff
@[to_additive]
theorem tendsto_const_smul_iff {f : β → α} {l : Filter β} {a : α} (c : G) :
    Tendsto (fun x => c • f x) l (𝓝 <| c • a) ↔ Tendsto f l (𝓝 a) 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] {f : β → α} {l : Filter β} {a : α} (c : G)
Goal: Filter.Tendsto (fun x => c +ᵥ f x) l (𝓝 (c +ᵥ a)) ↔ Filter.Tendsto f l (𝓝 a)
-- tendsto_const_smul_iff
@[to_additive]
theorem tendsto_const_smul_iff {f : β → α} {l : Filter β} {a : α} (c : G) :
    Tendsto (fun x => c • f x) l (𝓝 <| c • a) ↔ Tendsto f l (𝓝 a) 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] {f : β → α} {l : Filter β} {a : α} (c : G)
Goal: Filter.Tendsto (fun x => c • f x) l (𝓝 (c • a)) ↔ Filter.Tendsto f l (𝓝 a)
-- continuousWithinAt_const_vadd_iff
@[to_additive]
theorem continuousWithinAt_const_smul_iff (c : G) :
    ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} (c : G)
Goal: ContinuousWithinAt (fun x => c +ᵥ f x) s b ↔ ContinuousWithinAt f s b
-- continuousWithinAt_const_smul_iff
@[to_additive]
theorem continuousWithinAt_const_smul_iff (c : G) :
    ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} (c : G)
Goal: ContinuousWithinAt (fun x => c • f x) s b ↔ ContinuousWithinAt f s b
-- continuousOn_const_smul_iff
@[to_additive]
theorem continuousOn_const_smul_iff (c : G) :
    ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] [TopologicalSpace β] {f : β → α} {s : Set β} (c : G)
Goal: ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s
-- continuousOn_const_vadd_iff
@[to_additive]
theorem continuousOn_const_smul_iff (c : G) :
    ContinuousOn (fun x => c • f x) s ↔ ContinuousOn f s 
Arguments: {α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] [TopologicalSpace β] {f : β → α} {s : Set β} (c : G)
Goal: ContinuousOn (fun x => c +ᵥ f x) s ↔ ContinuousOn f s