ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TrivSqZeroExt.nhds_def
theorem nhds_def (x : tsze R M) : 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] (x : tsze R M)
Goal: 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd)
-- TrivSqZeroExt.nhds_inl
theorem nhds_inl [Zero M] (x : R) : 𝓝 (inl x : tsze R M) = (𝓝 x).prod (𝓝 0) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M] (x : R)
Goal: 𝓝 (TrivSqZeroExt.inl x) = (𝓝 x).prod (𝓝 0)
-- TrivSqZeroExt.nhds_inr
theorem nhds_inr [Zero R] (m : M) : 𝓝 (inr m : tsze R M) = (𝓝 0).prod (𝓝 m) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R] (m : M)
Goal: 𝓝 (TrivSqZeroExt.inr m) = (𝓝 0).prod (𝓝 m)
-- TrivSqZeroExt.continuous_fst
nonrec theorem continuous_fst : Continuous (fst : tsze R M → R) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M]
Goal: Continuous TrivSqZeroExt.fst
-- TrivSqZeroExt.continuous_snd
nonrec theorem continuous_snd : Continuous (snd : tsze R M → M) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M]
Goal: Continuous TrivSqZeroExt.snd
-- TrivSqZeroExt.continuous_inl
theorem continuous_inl [Zero M] : Continuous (inl : R → tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M]
Goal: Continuous TrivSqZeroExt.inl
-- TrivSqZeroExt.continuous_inr
theorem continuous_inr [Zero R] : Continuous (inr : M → tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R]
Goal: Continuous TrivSqZeroExt.inr
-- TrivSqZeroExt.embedding_inl
theorem embedding_inl [Zero M] : Embedding (inl : R → tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero M]
Goal: Embedding TrivSqZeroExt.inl
-- TrivSqZeroExt.embedding_inr
theorem embedding_inr [Zero R] : Embedding (inr : M → tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Zero R]
Goal: Embedding TrivSqZeroExt.inr
-- TrivSqZeroExt.fstCLM_apply
@[simps]
def fstCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] R 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (x : tsze R M)
Goal: (TrivSqZeroExt.fstCLM R M) x = x.fst
-- TrivSqZeroExt.fstCLM_toFun
@[simps]
def fstCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] R 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (x : tsze R M)
Goal: (TrivSqZeroExt.fstCLM R M) x = x.fst
-- TrivSqZeroExt.sndCLM_toFun
@[simps]
def sndCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] M 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (x : tsze R M)
Goal: (TrivSqZeroExt.sndCLM R M) x = x.snd
-- TrivSqZeroExt.sndCLM_apply
@[simps]
def sndCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] M 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (x : tsze R M)
Goal: (TrivSqZeroExt.sndCLM R M) x = x.snd
-- TrivSqZeroExt.inlCLM_toFun
@[simps]
def inlCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] tsze R M 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (r : R)
Goal: (TrivSqZeroExt.inlCLM R M) r = TrivSqZeroExt.inl r
-- TrivSqZeroExt.inlCLM_apply
@[simps]
def inlCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] tsze R M 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (r : R)
Goal: (TrivSqZeroExt.inlCLM R M) r = TrivSqZeroExt.inl r
-- TrivSqZeroExt.inrCLM_apply
@[simps]
def inrCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] tsze R M 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (m : M)
Goal: (TrivSqZeroExt.inrCLM R M) m = TrivSqZeroExt.inr m
-- TrivSqZeroExt.inrCLM_toFun
@[simps]
def inrCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] tsze R M 
Arguments: (R : Type u_3) (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [CommSemiring R] [AddCommMonoid M] [Module R M] (m : M)
Goal: (TrivSqZeroExt.inrCLM R M) m = TrivSqZeroExt.inr m
-- TrivSqZeroExt.topologicalSemiring
theorem topologicalSemiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
    [TopologicalSemiring R] [ContinuousAdd M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] :
    TopologicalSemiring (tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [TopologicalSpace R] [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [TopologicalSemiring R] [ContinuousAdd M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
Goal: TopologicalSemiring (tsze R M)
-- TrivSqZeroExt.hasSum_inl
theorem hasSum_inl [AddCommMonoid R] [AddCommMonoid M] {f : α → R} {a : R} (h : HasSum f a) :
    HasSum (fun x ↦ inl (f x)) (inl a : tsze R M) 
Arguments: {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → R} {a : R} (h : HasSum f a)
Goal: HasSum (fun x => TrivSqZeroExt.inl (f x)) (TrivSqZeroExt.inl a)
-- TrivSqZeroExt.hasSum_inr
theorem hasSum_inr [AddCommMonoid R] [AddCommMonoid M] {f : α → M} {a : M} (h : HasSum f a) :
    HasSum (fun x ↦ inr (f x)) (inr a : tsze R M) 
Arguments: {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → M} {a : M} (h : HasSum f a)
Goal: HasSum (fun x => TrivSqZeroExt.inr (f x)) (TrivSqZeroExt.inr a)
-- TrivSqZeroExt.hasSum_fst
theorem hasSum_fst [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M}
    (h : HasSum f a) : HasSum (fun x ↦ fst (f x)) (fst a) 
Arguments: {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M} (h : HasSum f a)
Goal: HasSum (fun x => (f x).fst) a.fst
-- TrivSqZeroExt.hasSum_snd
theorem hasSum_snd [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M}
    (h : HasSum f a) : HasSum (fun x ↦ snd (f x)) (snd a) 
Arguments: {α : Type u_1} {R : Type u_3} (M : Type u_4) [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M} (h : HasSum f a)
Goal: HasSum (fun x => (f x).snd) a.snd
-- TrivSqZeroExt.uniformity_def
theorem uniformity_def :
    𝓤 (tsze R M) =
      ((𝓤 R).comap fun p => (p.1.fst, p.2.fst)) ⊓ ((𝓤 M).comap fun p => (p.1.snd, p.2.snd)) 
Arguments: {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M]
Goal: uniformity (tsze R M) =
  Filter.comap (fun p => (p.1.fst, p.2.fst)) (uniformity R) ⊓ Filter.comap (fun p => (p.1.snd, p.2.snd)) (uniformity M)
-- TrivSqZeroExt.uniformContinuous_fst
nonrec theorem uniformContinuous_fst : UniformContinuous (fst : tsze R M → R) 
Arguments: {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M]
Goal: UniformContinuous TrivSqZeroExt.fst
-- TrivSqZeroExt.uniformContinuous_snd
nonrec theorem uniformContinuous_snd : UniformContinuous (snd : tsze R M → M) 
Arguments: {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M]
Goal: UniformContinuous TrivSqZeroExt.snd
-- TrivSqZeroExt.uniformContinuous_inl
theorem uniformContinuous_inl [Zero M] : UniformContinuous (inl : R → tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] [Zero M]
Goal: UniformContinuous TrivSqZeroExt.inl
-- TrivSqZeroExt.uniformContinuous_inr
theorem uniformContinuous_inr [Zero R] : UniformContinuous (inr : M → tsze R M) 
Arguments: {R : Type u_3} {M : Type u_4} [UniformSpace R] [UniformSpace M] [Zero R]
Goal: UniformContinuous TrivSqZeroExt.inr
-- Complex.subfield_eq_of_closed
theorem Complex.subfield_eq_of_closed {K : Subfield ℂ} (hc : IsClosed (K : Set ℂ)) :
    K = ofReal.fieldRange ∨ K = ⊤ 
Arguments: {K : Subfield ℂ} (hc : IsClosed ↑K)
Goal: K = Complex.ofReal.fieldRange ∨ K = ⊤
-- Complex.uniformContinuous_ringHom_eq_id_or_conj
theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
    (hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype 
Arguments: (K : Subfield ℂ) {ψ : ↥K →+* ℂ} (hc : UniformContinuous ⇑ψ)
Goal: (↑↑ψ).toFun = ⇑K.subtype ∨ (↑↑ψ).toFun = ⇑(starRingEnd ℂ) ∘ ⇑K.subtype
-- exists_compact_superset
theorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :
    ∃ K', IsCompact K' ∧ K ⊆ interior K' 
Arguments: {X : Type u_1} [TopologicalSpace X] [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K)
Goal: ∃ K', IsCompact K' ∧ K ⊆ interior K'
-- disjoint_nhds_cocompact
theorem disjoint_nhds_cocompact [WeaklyLocallyCompactSpace X] (x : X) :
    Disjoint (𝓝 x) (cocompact X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [WeaklyLocallyCompactSpace X] (x : X)
Goal: Disjoint (𝓝 x) (Filter.cocompact X)
-- compact_basis_nhds
theorem compact_basis_nhds [LocallyCompactSpace X] (x : X) :
    (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] (x : X)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s
-- local_compact_nhds
theorem local_compact_nhds [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ 𝓝 x) :
    ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ 𝓝 x)
Goal: ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s
-- LocallyCompactSpace.of_hasBasis
theorem LocallyCompactSpace.of_hasBasis {ι : X → Type*} {p : ∀ x, ι x → Prop}
    {s : ∀ x, ι x → Set X} (h : ∀ x, (𝓝 x).HasBasis (p x) (s x))
    (hc : ∀ x i, p x i → IsCompact (s x i)) : LocallyCompactSpace X 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : X → Type u_4} {p : (x : X) → ι x → Prop} {s : (x : X) → ι x → Set X} (h : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)) (hc : ∀ (x : X) (i : ι x), p x i → IsCompact (s x i))
Goal: LocallyCompactSpace X
-- exists_compact_subset
theorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)
    (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U) (hx : x ∈ U)
Goal: ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U
-- exists_mem_nhdsSet_isCompact_mapsTo
lemma exists_mem_nhdsSet_isCompact_mapsTo [LocallyCompactPair X Y] {f : X → Y} {K : Set X}
    {U : Set Y} (hf : Continuous f) (hK : IsCompact K) (hU : IsOpen U) (hKU : MapsTo f K U) :
    ∃ L ∈ 𝓝ˢ K, IsCompact L ∧ MapsTo f L U 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactPair X Y] {f : X → Y} {K : Set X} {U : Set Y} (hf : Continuous f) (hK : IsCompact K) (hU : IsOpen U) (hKU : Set.MapsTo f K U)
Goal: ∃ L ∈ 𝓝ˢ K, IsCompact L ∧ Set.MapsTo f L U
-- exists_compact_between
theorem exists_compact_between [LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K)
    (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U)
Goal: ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U
-- ClosedEmbedding.locallyCompactSpace
protected theorem ClosedEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
    (hf : ClosedEmbedding f) : LocallyCompactSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: LocallyCompactSpace X
-- IsClosed.locallyCompactSpace
protected theorem IsClosed.locallyCompactSpace [LocallyCompactSpace X] {s : Set X}
    (hs : IsClosed s) : LocallyCompactSpace s 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {s : Set X} (hs : IsClosed s)
Goal: LocallyCompactSpace ↑s
-- OpenEmbedding.locallyCompactSpace
protected theorem OpenEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
    (hf : OpenEmbedding f) : LocallyCompactSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace Y] {f : X → Y} (hf : OpenEmbedding f)
Goal: LocallyCompactSpace X
-- IsOpen.locallyCompactSpace
protected theorem IsOpen.locallyCompactSpace [LocallyCompactSpace X] {s : Set X} (hs : IsOpen s) :
    LocallyCompactSpace s 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {s : Set X} (hs : IsOpen s)
Goal: LocallyCompactSpace ↑s
-- quasiSeparatedSpace_iff
@[mk_iff]
class QuasiSeparatedSpace (α : Type*) [TopologicalSpace α] : Prop where
    inter_isCompact :
    ∀ U V : Set α, IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
Arguments: (α : Type u_3) [TopologicalSpace α]
Goal: QuasiSeparatedSpace α ↔ ∀ (U V : Set α), IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
-- QuasiSeparatedSpace.inter_isCompact
@[mk_iff]
class QuasiSeparatedSpace (α : Type*) [TopologicalSpace α] : Prop where
    inter_isCompact :
    ∀ U V : Set α, IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
Arguments: {α : Type u_3} [TopologicalSpace α] [self : QuasiSeparatedSpace α] (U V : Set α)
Goal: IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
-- isQuasiSeparated_univ_iff
theorem isQuasiSeparated_univ_iff {α : Type*} [TopologicalSpace α] :
    IsQuasiSeparated (Set.univ : Set α) ↔ QuasiSeparatedSpace α 
Arguments: {α : Type u_3} [TopologicalSpace α]
Goal: IsQuasiSeparated Set.univ ↔ QuasiSeparatedSpace α
-- isQuasiSeparated_univ
theorem isQuasiSeparated_univ {α : Type*} [TopologicalSpace α] [QuasiSeparatedSpace α] :
    IsQuasiSeparated (Set.univ : Set α) 
Arguments: {α : Type u_3} [TopologicalSpace α] [QuasiSeparatedSpace α]
Goal: IsQuasiSeparated Set.univ
-- IsQuasiSeparated.image_of_embedding
theorem IsQuasiSeparated.image_of_embedding {s : Set α} (H : IsQuasiSeparated s) (h : Embedding f) :
    IsQuasiSeparated (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (H : IsQuasiSeparated s) (h : Embedding f)
Goal: IsQuasiSeparated (f '' s)
-- OpenEmbedding.isQuasiSeparated_iff
theorem OpenEmbedding.isQuasiSeparated_iff (h : OpenEmbedding f) {s : Set α} :
    IsQuasiSeparated s ↔ IsQuasiSeparated (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (h : OpenEmbedding f) {s : Set α}
Goal: IsQuasiSeparated s ↔ IsQuasiSeparated (f '' s)
-- isQuasiSeparated_iff_quasiSeparatedSpace
theorem isQuasiSeparated_iff_quasiSeparatedSpace (s : Set α) (hs : IsOpen s) :
    IsQuasiSeparated s ↔ QuasiSeparatedSpace s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) (hs : IsOpen s)
Goal: IsQuasiSeparated s ↔ QuasiSeparatedSpace ↑s
-- IsQuasiSeparated.of_subset
theorem IsQuasiSeparated.of_subset {s t : Set α} (ht : IsQuasiSeparated t) (h : s ⊆ t) :
    IsQuasiSeparated s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} (ht : IsQuasiSeparated t) (h : s ⊆ t)
Goal: IsQuasiSeparated s
-- IsQuasiSeparated.of_quasiSeparatedSpace
theorem IsQuasiSeparated.of_quasiSeparatedSpace (s : Set α) [QuasiSeparatedSpace α] :
    IsQuasiSeparated s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α) [QuasiSeparatedSpace α]
Goal: IsQuasiSeparated s
-- QuasiSeparatedSpace.of_openEmbedding
theorem QuasiSeparatedSpace.of_openEmbedding (h : OpenEmbedding f) [QuasiSeparatedSpace β] :
    QuasiSeparatedSpace α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (h : OpenEmbedding f) [QuasiSeparatedSpace β]
Goal: QuasiSeparatedSpace α
-- IsClopen.isOpen
protected theorem IsClopen.isOpen (hs : IsClopen s) : IsOpen s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsClopen s)
Goal: IsOpen s
-- IsClopen.isClosed
protected theorem IsClopen.isClosed (hs : IsClopen s) : IsClosed s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsClopen s)
Goal: IsClosed s
-- isClopen_iff_frontier_eq_empty
theorem isClopen_iff_frontier_eq_empty : IsClopen s ↔ frontier s = ∅ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsClopen s ↔ frontier s = ∅
-- IsClopen.union
theorem IsClopen.union (hs : IsClopen s) (ht : IsClopen t) : IsClopen (s ∪ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsClopen s) (ht : IsClopen t)
Goal: IsClopen (s ∪ t)
-- IsClopen.inter
theorem IsClopen.inter (hs : IsClopen s) (ht : IsClopen t) : IsClopen (s ∩ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsClopen s) (ht : IsClopen t)
Goal: IsClopen (s ∩ t)
-- isClopen_empty
theorem isClopen_empty : IsClopen (∅ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClopen ∅
-- isClopen_univ
theorem isClopen_univ : IsClopen (univ : Set X) 
Arguments: {X : Type u} [TopologicalSpace X]
Goal: IsClopen Set.univ
-- IsClopen.compl
theorem IsClopen.compl (hs : IsClopen s) : IsClopen sᶜ 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsClopen s)
Goal: IsClopen sᶜ
-- isClopen_compl_iff
@[simp]
theorem isClopen_compl_iff : IsClopen sᶜ ↔ IsClopen s 
Arguments: {X : Type u} [TopologicalSpace X] {s : Set X}
Goal: IsClopen sᶜ ↔ IsClopen s
-- IsClopen.diff
theorem IsClopen.diff (hs : IsClopen s) (ht : IsClopen t) : IsClopen (s \ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsClopen s) (ht : IsClopen t)
Goal: IsClopen (s \ t)
-- IsClopen.himp
lemma IsClopen.himp (hs : IsClopen s) (ht : IsClopen t) : IsClopen (s ⇨ t) 
Arguments: {X : Type u} [TopologicalSpace X] {s t : Set X} (hs : IsClopen s) (ht : IsClopen t)
Goal: IsClopen (s ⇨ t)
-- IsClopen.prod
theorem IsClopen.prod {t : Set Y} (hs : IsClopen s) (ht : IsClopen t) : IsClopen (s ×ˢ t) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {t : Set Y} (hs : IsClopen s) (ht : IsClopen t)
Goal: IsClopen (s ×ˢ t)
-- isClopen_iUnion_of_finite
theorem isClopen_iUnion_of_finite [Finite Y] {s : Y → Set X} (h : ∀ i, IsClopen (s i)) :
    IsClopen (⋃ i, s i) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [Finite Y] {s : Y → Set X} (h : ∀ (i : Y), IsClopen (s i))
Goal: IsClopen (⋃ i, s i)
-- Set.Finite.isClopen_biUnion
theorem Set.Finite.isClopen_biUnion {s : Set Y} {f : Y → Set X} (hs : s.Finite)
    (h : ∀ i ∈ s, IsClopen <| f i) : IsClopen (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {s : Set Y} {f : Y → Set X} (hs : s.Finite) (h : ∀ i ∈ s, IsClopen (f i))
Goal: IsClopen (⋃ i ∈ s, f i)
-- isClopen_biUnion_finset
theorem isClopen_biUnion_finset {s : Finset Y} {f : Y → Set X}
    (h : ∀ i ∈ s, IsClopen <| f i) : IsClopen (⋃ i ∈ s, f i) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {s : Finset Y} {f : Y → Set X} (h : ∀ i ∈ s, IsClopen (f i))
Goal: IsClopen (⋃ i ∈ s, f i)
-- isClopen_iInter_of_finite
theorem isClopen_iInter_of_finite [Finite Y] {s : Y → Set X} (h : ∀ i, IsClopen (s i)) :
    IsClopen (⋂ i, s i) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [Finite Y] {s : Y → Set X} (h : ∀ (i : Y), IsClopen (s i))
Goal: IsClopen (⋂ i, s i)
-- Set.Finite.isClopen_biInter
theorem Set.Finite.isClopen_biInter {s : Set Y} (hs : s.Finite) {f : Y → Set X}
    (h : ∀ i ∈ s, IsClopen (f i)) : IsClopen (⋂ i ∈ s, f i) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {s : Set Y} (hs : s.Finite) {f : Y → Set X} (h : ∀ i ∈ s, IsClopen (f i))
Goal: IsClopen (⋂ i ∈ s, f i)
-- isClopen_biInter_finset
theorem isClopen_biInter_finset {s : Finset Y} {f : Y → Set X}
    (h : ∀ i ∈ s, IsClopen (f i)) : IsClopen (⋂ i ∈ s, f i) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] {s : Finset Y} {f : Y → Set X} (h : ∀ i ∈ s, IsClopen (f i))
Goal: IsClopen (⋂ i ∈ s, f i)
-- IsClopen.preimage
theorem IsClopen.preimage {s : Set Y} (h : IsClopen s) {f : X → Y} (hf : Continuous f) :
    IsClopen (f ⁻¹' s) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (h : IsClopen s) {f : X → Y} (hf : Continuous f)
Goal: IsClopen (f ⁻¹' s)
-- ContinuousOn.preimage_isClopen_of_isClopen
theorem ContinuousOn.preimage_isClopen_of_isClopen {f : X → Y} {s : Set X} {t : Set Y}
    (hf : ContinuousOn f s) (hs : IsClopen s) (ht : IsClopen t) : IsClopen (s ∩ f ⁻¹' t) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {t : Set Y} (hf : ContinuousOn f s) (hs : IsClopen s) (ht : IsClopen t)
Goal: IsClopen (s ∩ f ⁻¹' t)
-- isClopen_inter_of_disjoint_cover_clopen
theorem isClopen_inter_of_disjoint_cover_clopen {s a b : Set X} (h : IsClopen s) (cover : s ⊆ a ∪ b)
    (ha : IsOpen a) (hb : IsOpen b) (hab : Disjoint a b) : IsClopen (s ∩ a) 
Arguments: {X : Type u} [TopologicalSpace X] {s a b : Set X} (h : IsClopen s) (cover : s ⊆ a ∪ b) (ha : IsOpen a) (hb : IsOpen b) (hab : Disjoint a b)
Goal: IsClopen (s ∩ a)
-- isClopen_discrete
@[simp]
theorem isClopen_discrete [DiscreteTopology X] (s : Set X) : IsClopen s 
Arguments: {X : Type u} [TopologicalSpace X] [DiscreteTopology X] (s : Set X)
Goal: IsClopen s
-- isClopen_range_inl
theorem isClopen_range_inl : IsClopen (range (Sum.inl : X → X ⊕ Y)) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y]
Goal: IsClopen (Set.range Sum.inl)
-- isClopen_range_inr
theorem isClopen_range_inr : IsClopen (range (Sum.inr : Y → X ⊕ Y)) 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y]
Goal: IsClopen (Set.range Sum.inr)
-- isClopen_range_sigmaMk
theorem isClopen_range_sigmaMk {X : ι → Type*} [∀ i, TopologicalSpace (X i)] {i : ι} :
    IsClopen (Set.range (@Sigma.mk ι X i)) 
Arguments: {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {i : ι}
Goal: IsClopen (Set.range (Sigma.mk i))
-- QuotientMap.isClopen_preimage
protected theorem QuotientMap.isClopen_preimage {f : X → Y} (hf : QuotientMap f) {s : Set Y} :
    IsClopen (f ⁻¹' s) ↔ IsClopen s 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : QuotientMap f) {s : Set Y}
Goal: IsClopen (f ⁻¹' s) ↔ IsClopen s
-- continuous_boolIndicator_iff_isClopen
theorem continuous_boolIndicator_iff_isClopen (U : Set X) :
    Continuous U.boolIndicator ↔ IsClopen U 
Arguments: {X : Type u} [TopologicalSpace X] (U : Set X)
Goal: Continuous U.boolIndicator ↔ IsClopen U
-- continuousOn_boolIndicator_iff_isClopen
theorem continuousOn_boolIndicator_iff_isClopen (s U : Set X) :
    ContinuousOn U.boolIndicator s ↔ IsClopen (((↑) : s → X) ⁻¹' U) 
Arguments: {X : Type u} [TopologicalSpace X] (s U : Set X)
Goal: ContinuousOn U.boolIndicator s ↔ IsClopen (Subtype.val ⁻¹' U)
-- ContinuousMap.Homotopy.map_zero_left
structure Homotopy (f₀ f₁ : C(X, Y)) extends C(I × X, Y) where
    map_zero_left : ∀ x, toFun (0, x) = f₀ x
    map_one_left : ∀ x, toFun (1, x) = f₁ x
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f₀ f₁ : C(X, Y)} (self : f₀.Homotopy f₁) (x : X)
Goal: self.toFun (0, x) = f₀ x
-- ContinuousMap.Homotopy.map_one_left
structure Homotopy (f₀ f₁ : C(X, Y)) extends C(I × X, Y) where
    map_zero_left : ∀ x, toFun (0, x) = f₀ x
    map_one_left : ∀ x, toFun (1, x) = f₁ x
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f₀ f₁ : C(X, Y)} (self : f₀.Homotopy f₁) (x : X)
Goal: self.toFun (1, x) = f₁ x
-- ContinuousMap.HomotopyLike.map_zero_left
class HomotopyLike {X Y : outParam Type*} [TopologicalSpace X] [TopologicalSpace Y]
    (F : Type*) (f₀ f₁ : outParam <| C(X, Y)) [FunLike F (I × X) Y]
    extends ContinuousMapClass F (I × X) Y : Prop where
    map_zero_left (f : F) : ∀ x, f (0, x) = f₀ x
    map_one_left (f : F) : ∀ x, f (1, x) = f₁ x
Arguments: {X : outParam (Type u_3)} {Y : outParam (Type u_4)} [TopologicalSpace X] [TopologicalSpace Y] {F : Type u_5} {f₀ f₁ : outParam C(X, Y)} [FunLike F (↑I × X) Y] [self : ContinuousMap.HomotopyLike F f₀ f₁] (f : F) (x : X)
Goal: f (0, x) = f₀ x