ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ModelWithCorners.disjoint_interior_boundary
lemma disjoint_interior_boundary : Disjoint (I.interior M) (I.boundary M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M]
Goal: Disjoint (I.interior M) (I.boundary M)
-- ModelWithCorners.boundary_eq_complement_interior
lemma boundary_eq_complement_interior : I.boundary M = (I.interior M)ᶜ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M]
Goal: I.boundary M = (I.interior M)ᶜ
-- range_mem_nhds_isInteriorPoint
lemma _root_.range_mem_nhds_isInteriorPoint {x : M} (h : I.IsInteriorPoint x) :
    range I ∈ 𝓝 (extChartAt I x x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {x : M} (h : I.IsInteriorPoint x)
Goal: Set.range ↑I ∈ 𝓝 (↑(extChartAt I x) x)
-- BoundarylessManifold.isInteriorPoint'
class _root_.BoundarylessManifold {𝕜 : Type*} [NontriviallyNormedField 𝕜]
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)
    (M : Type*) [TopologicalSpace M] [ChartedSpace H M] : Prop where
  isInteriorPoint' : ∀ x : M, IsInteriorPoint I x
Arguments: {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] {E : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_7} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H M] [self : BoundarylessManifold I M] (x : M)
Goal: I.IsInteriorPoint x
-- BoundarylessManifold.isInteriorPoint
lemma _root_.BoundarylessManifold.isInteriorPoint {x : M} :
    IsInteriorPoint I x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [BoundarylessManifold I M] {x : M}
Goal: I.IsInteriorPoint x
-- ModelWithCorners.interior_eq_univ
lemma interior_eq_univ : I.interior M = univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [BoundarylessManifold I M]
Goal: I.interior M = Set.univ
-- ModelWithCorners.Boundaryless.boundary_eq_empty
lemma Boundaryless.boundary_eq_empty : I.boundary M = ∅ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [BoundarylessManifold I M]
Goal: I.boundary M = ∅
-- ModelWithCorners.interior_prod
lemma ModelWithCorners.interior_prod :
    (I.prod J).interior (M × N) = (I.interior M) ×ˢ (J.interior N) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {N : Type u_7} [TopologicalSpace N] [ChartedSpace H' N] (J : ModelWithCorners 𝕜 E' H')
Goal: (I.prod J).interior (M × N) = I.interior M ×ˢ J.interior N
-- ModelWithCorners.boundary_prod
lemma ModelWithCorners.boundary_prod :
    (I.prod J).boundary (M × N) = Set.prod univ (J.boundary N) ∪ Set.prod (I.boundary M) univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {N : Type u_7} [TopologicalSpace N] [ChartedSpace H' N] (J : ModelWithCorners 𝕜 E' H')
Goal: (I.prod J).boundary (M × N) = Set.univ.prod (J.boundary N) ∪ (I.boundary M).prod Set.univ
-- boundary_of_boundaryless_left
lemma boundary_of_boundaryless_left [I.Boundaryless] :
    (I.prod J).boundary (M × N) = Set.prod (univ : Set M) (J.boundary N) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {N : Type u_7} [TopologicalSpace N] [ChartedSpace H' N] (J : ModelWithCorners 𝕜 E' H') [I.Boundaryless]
Goal: (I.prod J).boundary (M × N) = Set.univ.prod (J.boundary N)
-- boundary_of_boundaryless_right
lemma boundary_of_boundaryless_right [J.Boundaryless] :
    (I.prod J).boundary (M × N) = Set.prod (I.boundary M) (univ : Set N) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {N : Type u_7} [TopologicalSpace N] [ChartedSpace H' N] (J : ModelWithCorners 𝕜 E' H') [J.Boundaryless]
Goal: (I.prod J).boundary (M × N) = (I.boundary M).prod Set.univ
-- Units.chartAt_apply
theorem chartAt_apply {a : Rˣ} {b : Rˣ} : chartAt R a b = b 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] {a b : Rˣ}
Goal: ↑(chartAt R a) b = ↑b
-- Units.chartAt_source
theorem chartAt_source {a : Rˣ} : (chartAt R a).source = Set.univ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] {a : Rˣ}
Goal: (chartAt R a).source = Set.univ
-- Units.contMDiff_val
lemma contMDiff_val {m : ℕ∞} : ContMDiff 𝓘(𝕜, R) 𝓘(𝕜, R) m (val : Rˣ → R) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 R] {m : ℕ∞}
Goal: ContMDiff 𝓘(𝕜, R) 𝓘(𝕜, R) m Units.val
-- IsConnected.nonempty
theorem IsConnected.nonempty {s : Set α} (h : IsConnected s) : s.Nonempty 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (h : IsConnected s)
Goal: s.Nonempty
-- IsConnected.isPreconnected
theorem IsConnected.isPreconnected {s : Set α} (h : IsConnected s) : IsPreconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (h : IsConnected s)
Goal: IsPreconnected s
-- IsPreirreducible.isPreconnected
theorem IsPreirreducible.isPreconnected {s : Set α} (H : IsPreirreducible s) : IsPreconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (H : IsPreirreducible s)
Goal: IsPreconnected s
-- IsIrreducible.isConnected
theorem IsIrreducible.isConnected {s : Set α} (H : IsIrreducible s) : IsConnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (H : IsIrreducible s)
Goal: IsConnected s
-- isPreconnected_empty
theorem isPreconnected_empty : IsPreconnected (∅ : Set α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: IsPreconnected ∅
-- isConnected_singleton
theorem isConnected_singleton {x} : IsConnected ({x} : Set α) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsConnected {x}
-- isPreconnected_singleton
theorem isPreconnected_singleton {x} : IsPreconnected ({x} : Set α) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsPreconnected {x}
-- Set.Subsingleton.isPreconnected
theorem Set.Subsingleton.isPreconnected {s : Set α} (hs : s.Subsingleton) : IsPreconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (hs : s.Subsingleton)
Goal: IsPreconnected s
-- isPreconnected_of_forall
theorem isPreconnected_of_forall {s : Set α} (x : α)
    (H : ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) : IsPreconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (x : α) (H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t)
Goal: IsPreconnected s
-- isPreconnected_of_forall_pair
theorem isPreconnected_of_forall_pair {s : Set α}
    (H : ∀ x ∈ s, ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) :
    IsPreconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (H : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t)
Goal: IsPreconnected s
-- isPreconnected_sUnion
theorem isPreconnected_sUnion (x : α) (c : Set (Set α)) (H1 : ∀ s ∈ c, x ∈ s)
    (H2 : ∀ s ∈ c, IsPreconnected s) : IsPreconnected (⋃₀ c) 
Arguments: {α : Type u} [TopologicalSpace α] (x : α) (c : Set (Set α)) (H1 : ∀ s ∈ c, x ∈ s) (H2 : ∀ s ∈ c, IsPreconnected s)
Goal: IsPreconnected (⋃₀ c)
-- isPreconnected_iUnion
theorem isPreconnected_iUnion {ι : Sort*} {s : ι → Set α} (h₁ : (⋂ i, s i).Nonempty)
    (h₂ : ∀ i, IsPreconnected (s i)) : IsPreconnected (⋃ i, s i) 
Arguments: {α : Type u} [TopologicalSpace α] {ι : Sort u_3} {s : ι → Set α} (h₁ : (⋂ i, s i).Nonempty) (h₂ : ∀ (i : ι), IsPreconnected (s i))
Goal: IsPreconnected (⋃ i, s i)
-- IsPreconnected.union
theorem IsPreconnected.union (x : α) {s t : Set α} (H1 : x ∈ s) (H2 : x ∈ t) (H3 : IsPreconnected s)
    (H4 : IsPreconnected t) : IsPreconnected (s ∪ t) 
Arguments: {α : Type u} [TopologicalSpace α] (x : α) {s t : Set α} (H1 : x ∈ s) (H2 : x ∈ t) (H3 : IsPreconnected s) (H4 : IsPreconnected t)
Goal: IsPreconnected (s ∪ t)
-- IsPreconnected.union'
theorem IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
    (ht : IsPreconnected t) : IsPreconnected (s ∪ t) 
Arguments: {α : Type u} [TopologicalSpace α] {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s) (ht : IsPreconnected t)
Goal: IsPreconnected (s ∪ t)
-- IsConnected.union
theorem IsConnected.union {s t : Set α} (H : (s ∩ t).Nonempty) (Hs : IsConnected s)
    (Ht : IsConnected t) : IsConnected (s ∪ t) 
Arguments: {α : Type u} [TopologicalSpace α] {s t : Set α} (H : (s ∩ t).Nonempty) (Hs : IsConnected s) (Ht : IsConnected t)
Goal: IsConnected (s ∪ t)
-- IsPreconnected.sUnion_directed
theorem IsPreconnected.sUnion_directed {S : Set (Set α)} (K : DirectedOn (· ⊆ ·) S)
    (H : ∀ s ∈ S, IsPreconnected s) : IsPreconnected (⋃₀ S) 
Arguments: {α : Type u} [TopologicalSpace α] {S : Set (Set α)} (K : DirectedOn (fun x x_1 => x ⊆ x_1) S) (H : ∀ s ∈ S, IsPreconnected s)
Goal: IsPreconnected (⋃₀ S)
-- IsPreconnected.biUnion_of_reflTransGen
theorem IsPreconnected.biUnion_of_reflTransGen {ι : Type*} {t : Set ι} {s : ι → Set α}
    (H : ∀ i ∈ t, IsPreconnected (s i))
    (K : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j) :
    IsPreconnected (⋃ n ∈ t, s n) 
Arguments: {α : Type u} [TopologicalSpace α] {ι : Type u_3} {t : Set ι} {s : ι → Set α} (H : ∀ i ∈ t, IsPreconnected (s i)) (K : ∀ i ∈ t, ∀ j ∈ t, Relation.ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j)
Goal: IsPreconnected (⋃ n ∈ t, s n)
-- IsConnected.biUnion_of_reflTransGen
theorem IsConnected.biUnion_of_reflTransGen {ι : Type*} {t : Set ι} {s : ι → Set α}
    (ht : t.Nonempty) (H : ∀ i ∈ t, IsConnected (s i))
    (K : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty ∧ i ∈ t) i j) :
    IsConnected (⋃ n ∈ t, s n) 
Arguments: {α : Type u} [TopologicalSpace α] {ι : Type u_3} {t : Set ι} {s : ι → Set α} (ht : t.Nonempty) (H : ∀ i ∈ t, IsConnected (s i)) (K : ∀ i ∈ t, ∀ j ∈ t, Relation.ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j)
Goal: IsConnected (⋃ n ∈ t, s n)
-- IsPreconnected.iUnion_of_reflTransGen
theorem IsPreconnected.iUnion_of_reflTransGen {ι : Type*} {s : ι → Set α}
    (H : ∀ i, IsPreconnected (s i))
    (K : ∀ i j, ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty) i j) :
    IsPreconnected (⋃ n, s n) 
Arguments: {α : Type u} [TopologicalSpace α] {ι : Type u_3} {s : ι → Set α} (H : ∀ (i : ι), IsPreconnected (s i)) (K : ∀ (i j : ι), Relation.ReflTransGen (fun i j => (s i ∩ s j).Nonempty) i j)
Goal: IsPreconnected (⋃ n, s n)
-- IsConnected.iUnion_of_reflTransGen
theorem IsConnected.iUnion_of_reflTransGen {ι : Type*} [Nonempty ι] {s : ι → Set α}
    (H : ∀ i, IsConnected (s i))
    (K : ∀ i j, ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty) i j) : IsConnected (⋃ n, s n) 
Arguments: {α : Type u} [TopologicalSpace α] {ι : Type u_3} [Nonempty ι] {s : ι → Set α} (H : ∀ (i : ι), IsConnected (s i)) (K : ∀ (i j : ι), Relation.ReflTransGen (fun i j => (s i ∩ s j).Nonempty) i j)
Goal: IsConnected (⋃ n, s n)
-- IsPreconnected.iUnion_of_chain
theorem IsPreconnected.iUnion_of_chain {s : β → Set α} (H : ∀ n, IsPreconnected (s n))
    (K : ∀ n, (s n ∩ s (succ n)).Nonempty) : IsPreconnected (⋃ n, s n) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder β] [SuccOrder β] [IsSuccArchimedean β] {s : β → Set α} (H : ∀ (n : β), IsPreconnected (s n)) (K : ∀ (n : β), (s n ∩ s (Order.succ n)).Nonempty)
Goal: IsPreconnected (⋃ n, s n)
-- IsConnected.iUnion_of_chain
theorem IsConnected.iUnion_of_chain [Nonempty β] {s : β → Set α} (H : ∀ n, IsConnected (s n))
    (K : ∀ n, (s n ∩ s (succ n)).Nonempty) : IsConnected (⋃ n, s n) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder β] [SuccOrder β] [IsSuccArchimedean β] [Nonempty β] {s : β → Set α} (H : ∀ (n : β), IsConnected (s n)) (K : ∀ (n : β), (s n ∩ s (Order.succ n)).Nonempty)
Goal: IsConnected (⋃ n, s n)
-- IsPreconnected.biUnion_of_chain
theorem IsPreconnected.biUnion_of_chain {s : β → Set α} {t : Set β} (ht : OrdConnected t)
    (H : ∀ n ∈ t, IsPreconnected (s n))
    (K : ∀ n : β, n ∈ t → succ n ∈ t → (s n ∩ s (succ n)).Nonempty) :
    IsPreconnected (⋃ n ∈ t, s n) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder β] [SuccOrder β] [IsSuccArchimedean β] {s : β → Set α} {t : Set β} (ht : t.OrdConnected) (H : ∀ n ∈ t, IsPreconnected (s n)) (K : ∀ n ∈ t, Order.succ n ∈ t → (s n ∩ s (Order.succ n)).Nonempty)
Goal: IsPreconnected (⋃ n ∈ t, s n)
-- IsConnected.biUnion_of_chain
theorem IsConnected.biUnion_of_chain {s : β → Set α} {t : Set β} (hnt : t.Nonempty)
    (ht : OrdConnected t) (H : ∀ n ∈ t, IsConnected (s n))
    (K : ∀ n : β, n ∈ t → succ n ∈ t → (s n ∩ s (succ n)).Nonempty) : IsConnected (⋃ n ∈ t, s n) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder β] [SuccOrder β] [IsSuccArchimedean β] {s : β → Set α} {t : Set β} (hnt : t.Nonempty) (ht : t.OrdConnected) (H : ∀ n ∈ t, IsConnected (s n)) (K : ∀ n ∈ t, Order.succ n ∈ t → (s n ∩ s (Order.succ n)).Nonempty)
Goal: IsConnected (⋃ n ∈ t, s n)
-- IsPreconnected.subset_closure
protected theorem IsPreconnected.subset_closure {s : Set α} {t : Set α} (H : IsPreconnected s)
    (Kst : s ⊆ t) (Ktcs : t ⊆ closure s) : IsPreconnected t 
Arguments: {α : Type u} [TopologicalSpace α] {s t : Set α} (H : IsPreconnected s) (Kst : s ⊆ t) (Ktcs : t ⊆ closure s)
Goal: IsPreconnected t
-- IsConnected.subset_closure
protected theorem IsConnected.subset_closure {s : Set α} {t : Set α} (H : IsConnected s)
    (Kst : s ⊆ t) (Ktcs : t ⊆ closure s) : IsConnected t 
Arguments: {α : Type u} [TopologicalSpace α] {s t : Set α} (H : IsConnected s) (Kst : s ⊆ t) (Ktcs : t ⊆ closure s)
Goal: IsConnected t
-- IsPreconnected.closure
protected theorem IsPreconnected.closure {s : Set α} (H : IsPreconnected s) :
    IsPreconnected (closure s) 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (H : IsPreconnected s)
Goal: IsPreconnected (closure s)
-- IsConnected.closure
protected theorem IsConnected.closure {s : Set α} (H : IsConnected s) : IsConnected (closure s) 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (H : IsConnected s)
Goal: IsConnected (closure s)
-- IsPreconnected.image
protected theorem IsPreconnected.image [TopologicalSpace β] {s : Set α} (H : IsPreconnected s)
    (f : α → β) (hf : ContinuousOn f s) : IsPreconnected (f '' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} (H : IsPreconnected s) (f : α → β) (hf : ContinuousOn f s)
Goal: IsPreconnected (f '' s)
-- IsConnected.image
protected theorem IsConnected.image [TopologicalSpace β] {s : Set α} (H : IsConnected s) (f : α → β)
    (hf : ContinuousOn f s) : IsConnected (f '' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} (H : IsConnected s) (f : α → β) (hf : ContinuousOn f s)
Goal: IsConnected (f '' s)
-- isPreconnected_closed_iff
theorem isPreconnected_closed_iff {s : Set α} :
    IsPreconnected s ↔ ∀ t t', IsClosed t → IsClosed t' →
      s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: IsPreconnected s ↔
  ∀ (t t' : Set α),
    IsClosed t → IsClosed t' → s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty
-- Inducing.isPreconnected_image
theorem Inducing.isPreconnected_image [TopologicalSpace β] {s : Set α} {f : α → β}
    (hf : Inducing f) : IsPreconnected (f '' s) ↔ IsPreconnected s 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : α → β} (hf : Inducing f)
Goal: IsPreconnected (f '' s) ↔ IsPreconnected s
-- IsPreconnected.preimage_of_isOpenMap
theorem IsPreconnected.preimage_of_isOpenMap [TopologicalSpace β] {f : α → β} {s : Set β}
    (hs : IsPreconnected s) (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ range f) :
    IsPreconnected (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set β} (hs : IsPreconnected s) (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ Set.range f)
Goal: IsPreconnected (f ⁻¹' s)
-- IsPreconnected.preimage_of_isClosedMap
theorem IsPreconnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β}
    (hs : IsPreconnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f)
    (hsf : s ⊆ range f) : IsPreconnected (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set β} (hs : IsPreconnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f) (hsf : s ⊆ Set.range f)
Goal: IsPreconnected (f ⁻¹' s)
-- IsConnected.preimage_of_isOpenMap
theorem IsConnected.preimage_of_isOpenMap [TopologicalSpace β] {s : Set β} (hs : IsConnected s)
    {f : α → β} (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ range f) :
    IsConnected (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set β} (hs : IsConnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ Set.range f)
Goal: IsConnected (f ⁻¹' s)
-- IsConnected.preimage_of_isClosedMap
theorem IsConnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β} (hs : IsConnected s)
    {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f) (hsf : s ⊆ range f) :
    IsConnected (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set β} (hs : IsConnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f) (hsf : s ⊆ Set.range f)
Goal: IsConnected (f ⁻¹' s)
-- IsPreconnected.subset_or_subset
theorem IsPreconnected.subset_or_subset (hu : IsOpen u) (hv : IsOpen v) (huv : Disjoint u v)
    (hsuv : s ⊆ u ∪ v) (hs : IsPreconnected s) : s ⊆ u ∨ s ⊆ v 
Arguments: {α : Type u} [TopologicalSpace α] {s u v : Set α} (hu : IsOpen u) (hv : IsOpen v) (huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hs : IsPreconnected s)
Goal: s ⊆ u ∨ s ⊆ v
-- IsPreconnected.subset_left_of_subset_union
theorem IsPreconnected.subset_left_of_subset_union (hu : IsOpen u) (hv : IsOpen v)
    (huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hsu : (s ∩ u).Nonempty) (hs : IsPreconnected s) :
    s ⊆ u 
Arguments: {α : Type u} [TopologicalSpace α] {s u v : Set α} (hu : IsOpen u) (hv : IsOpen v) (huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hsu : (s ∩ u).Nonempty) (hs : IsPreconnected s)
Goal: s ⊆ u
-- IsPreconnected.subset_right_of_subset_union
theorem IsPreconnected.subset_right_of_subset_union (hu : IsOpen u) (hv : IsOpen v)
    (huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hsv : (s ∩ v).Nonempty) (hs : IsPreconnected s) :
    s ⊆ v 
Arguments: {α : Type u} [TopologicalSpace α] {s u v : Set α} (hu : IsOpen u) (hv : IsOpen v) (huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hsv : (s ∩ v).Nonempty) (hs : IsPreconnected s)
Goal: s ⊆ v
-- IsPreconnected.subset_of_closure_inter_subset
theorem IsPreconnected.subset_of_closure_inter_subset (hs : IsPreconnected s) (hu : IsOpen u)
    (h'u : (s ∩ u).Nonempty) (h : closure u ∩ s ⊆ u) : s ⊆ u 
Arguments: {α : Type u} [TopologicalSpace α] {s u : Set α} (hs : IsPreconnected s) (hu : IsOpen u) (h'u : (s ∩ u).Nonempty) (h : closure u ∩ s ⊆ u)
Goal: s ⊆ u
-- IsPreconnected.prod
theorem IsPreconnected.prod [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsPreconnected s)
    (ht : IsPreconnected t) : IsPreconnected (s ×ˢ t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsPreconnected s) (ht : IsPreconnected t)
Goal: IsPreconnected (s ×ˢ t)
-- IsConnected.prod
theorem IsConnected.prod [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsConnected s)
    (ht : IsConnected t) : IsConnected (s ×ˢ t) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsConnected s) (ht : IsConnected t)
Goal: IsConnected (s ×ˢ t)
-- isPreconnected_univ_pi
theorem isPreconnected_univ_pi [∀ i, TopologicalSpace (π i)] {s : ∀ i, Set (π i)}
    (hs : ∀ i, IsPreconnected (s i)) : IsPreconnected (pi univ s) 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → TopologicalSpace (π i)] {s : (i : ι) → Set (π i)} (hs : ∀ (i : ι), IsPreconnected (s i))
Goal: IsPreconnected (Set.univ.pi s)
-- isConnected_univ_pi
@[simp]
theorem isConnected_univ_pi [∀ i, TopologicalSpace (π i)] {s : ∀ i, Set (π i)} :
    IsConnected (pi univ s) ↔ ∀ i, IsConnected (s i) 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → TopologicalSpace (π i)] {s : (i : ι) → Set (π i)}
Goal: IsConnected (Set.univ.pi s) ↔ ∀ (i : ι), IsConnected (s i)
-- connectedComponentIn_eq_image
theorem connectedComponentIn_eq_image {F : Set α} {x : α} (h : x ∈ F) :
    connectedComponentIn F x = (↑) '' connectedComponent (⟨x, h⟩ : F) 
Arguments: {α : Type u} [TopologicalSpace α] {F : Set α} {x : α} (h : x ∈ F)
Goal: connectedComponentIn F x = Subtype.val '' connectedComponent ⟨x, h⟩
-- connectedComponentIn_eq_empty
theorem connectedComponentIn_eq_empty {F : Set α} {x : α} (h : x ∉ F) :
    connectedComponentIn F x = ∅ 
Arguments: {α : Type u} [TopologicalSpace α] {F : Set α} {x : α} (h : x ∉ F)
Goal: connectedComponentIn F x = ∅
-- mem_connectedComponent
theorem mem_connectedComponent {x : α} : x ∈ connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: x ∈ connectedComponent x
-- mem_connectedComponentIn
theorem mem_connectedComponentIn {x : α} {F : Set α} (hx : x ∈ F) :
    x ∈ connectedComponentIn F x 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {F : Set α} (hx : x ∈ F)
Goal: x ∈ connectedComponentIn F x
-- connectedComponent_nonempty
theorem connectedComponent_nonempty {x : α} : (connectedComponent x).Nonempty 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: (connectedComponent x).Nonempty
-- connectedComponentIn_nonempty_iff
theorem connectedComponentIn_nonempty_iff {x : α} {F : Set α} :
    (connectedComponentIn F x).Nonempty ↔ x ∈ F 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {F : Set α}
Goal: (connectedComponentIn F x).Nonempty ↔ x ∈ F
-- connectedComponentIn_subset
theorem connectedComponentIn_subset (F : Set α) (x : α) : connectedComponentIn F x ⊆ F 
Arguments: {α : Type u} [TopologicalSpace α] (F : Set α) (x : α)
Goal: connectedComponentIn F x ⊆ F
-- isPreconnected_connectedComponent
theorem isPreconnected_connectedComponent {x : α} : IsPreconnected (connectedComponent x) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsPreconnected (connectedComponent x)
-- isPreconnected_connectedComponentIn
theorem isPreconnected_connectedComponentIn {x : α} {F : Set α} :
    IsPreconnected (connectedComponentIn F x) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {F : Set α}
Goal: IsPreconnected (connectedComponentIn F x)