ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- polynomialFunctions.comap_compRightAlgHom_iccHomeoI
theorem polynomialFunctions.comap_compRightAlgHom_iccHomeoI (a b : ℝ) (h : a < b) :
    (polynomialFunctions I).comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) =
      polynomialFunctions (Set.Icc a b) 
Arguments: (a b : ℝ) (h : a < b)
Goal: Subalgebra.comap (ContinuousMap.compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap)
    (polynomialFunctions unitInterval) =
  polynomialFunctions (Set.Icc a b)
-- polynomialFunctions.eq_adjoin_X
theorem polynomialFunctions.eq_adjoin_X (s : Set R) :
    polynomialFunctions s = Algebra.adjoin R {toContinuousMapOnAlgHom s X} 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] (s : Set R)
Goal: polynomialFunctions s = Algebra.adjoin R {(Polynomial.toContinuousMapOnAlgHom s) Polynomial.X}
-- polynomialFunctions.le_equalizer
theorem polynomialFunctions.le_equalizer {A : Type*} [Semiring A] [Algebra R A] (s : Set R)
    (φ ψ : C(s, R) →ₐ[R] A)
    (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) :
    polynomialFunctions s ≤ AlgHom.equalizer φ ψ 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] {A : Type u_2} [Semiring A] [Algebra R A] (s : Set R) (φ ψ : C(↑s, R) →ₐ[R] A) (h : φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X))
Goal: polynomialFunctions s ≤ AlgHom.equalizer φ ψ
-- polynomialFunctions.starClosure_eq_adjoin_X
theorem polynomialFunctions.starClosure_eq_adjoin_X [StarRing R] [ContinuousStar R] (s : Set R) :
    (polynomialFunctions s).starClosure = adjoin R {toContinuousMapOnAlgHom s X} 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] (s : Set R)
Goal: (polynomialFunctions s).starClosure = StarAlgebra.adjoin R {(Polynomial.toContinuousMapOnAlgHom s) Polynomial.X}
-- polynomialFunctions.starClosure_le_equalizer
theorem polynomialFunctions.starClosure_le_equalizer {A : Type*} [StarRing R] [ContinuousStar R]
    [Semiring A] [StarRing A] [Algebra R A] (s : Set R) (φ ψ : C(s, R) →⋆ₐ[R] A)
    (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) :
    (polynomialFunctions s).starClosure ≤ StarAlgHom.equalizer φ ψ 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] {A : Type u_2} [StarRing R] [ContinuousStar R] [Semiring A] [StarRing A] [Algebra R A] (s : Set R) (φ ψ : C(↑s, R) →⋆ₐ[R] A) (h : φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X))
Goal: (polynomialFunctions s).starClosure ≤ StarAlgHom.equalizer φ ψ
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.one_le_maxVar
private theorem one_le_maxVar : 1 ≤ maxVar X Y 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [MetricSpace Y]
Goal: 1 ≤ GromovHausdorff.maxVar X Y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.maxVar_bound
private theorem maxVar_bound : dist x y ≤ maxVar X Y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y] {x y : X ⊕ Y}
Goal: dist x y ≤ ↑(GromovHausdorff.maxVar X Y)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_symm
private theorem candidates_symm (fA : f ∈ candidates X Y) : f (x, y) = f (y, x) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} {x y : X ⊕ Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: f (x, y) = f (y, x)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_triangle
private theorem candidates_triangle (fA : f ∈ candidates X Y) : f (x, z) ≤ f (x, y) + f (y, z) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} {x y z : X ⊕ Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: f (x, z) ≤ f (x, y) + f (y, z)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_refl
private theorem candidates_refl (fA : f ∈ candidates X Y) : f (x, x) = 0 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} {x : X ⊕ Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: f (x, x) = 0
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_nonneg
private theorem candidates_nonneg (fA : f ∈ candidates X Y) : 0 ≤ f (x, y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} {x y : X ⊕ Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: 0 ≤ f (x, y)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_dist_inl
private theorem candidates_dist_inl (fA : f ∈ candidates X Y) (x y : X) :
    f (inl x, inl y) = dist x y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} (fA : f ∈ GromovHausdorff.candidates X Y) (x y : X)
Goal: f (Sum.inl x, Sum.inl y) = dist x y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_dist_inr
private theorem candidates_dist_inr (fA : f ∈ candidates X Y) (x y : Y) :
    f (inr x, inr y) = dist x y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} (fA : f ∈ GromovHausdorff.candidates X Y) (x y : Y)
Goal: f (Sum.inr x, Sum.inr y) = dist x y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_le_maxVar
private theorem candidates_le_maxVar (fA : f ∈ candidates X Y) : f (x, y) ≤ maxVar X Y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} {x y : X ⊕ Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_dist_bound
private theorem candidates_dist_bound (fA : f ∈ candidates X Y) :
    ∀ {x y : X ⊕ Y}, f (x, y) ≤ maxVar X Y * dist x y
  
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} (fA : f ∈ GromovHausdorff.candidates X Y) {x y : X ⊕ Y}
Goal: f (x, y) ≤ ↑(GromovHausdorff.maxVar X Y) * dist x y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_lipschitz_aux
private theorem candidates_lipschitz_aux (fA : f ∈ candidates X Y) :
    f (x, y) - f (z, t) ≤ 2 * maxVar X Y * dist (x, y) (z, t) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} {x y z t : X ⊕ Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: f (x, y) - f (z, t) ≤ 2 * ↑(GromovHausdorff.maxVar X Y) * dist (x, y) (z, t)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_lipschitz
private theorem candidates_lipschitz (fA : f ∈ candidates X Y) :
    LipschitzWith (2 * maxVar X Y) f 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun X Y} (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: LipschitzWith (2 * GromovHausdorff.maxVar X Y) f
-- GromovHausdorff.candidatesBOfCandidates_mem
theorem candidatesBOfCandidates_mem (f : ProdSpaceFun X Y) (fA : f ∈ candidates X Y) :
    candidatesBOfCandidates f fA ∈ candidatesB X Y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [MetricSpace Y] [CompactSpace Y] (f : GromovHausdorff.ProdSpaceFun X Y) (fA : f ∈ GromovHausdorff.candidates X Y)
Goal: GromovHausdorff.candidatesBOfCandidates f fA ∈ GromovHausdorff.candidatesB X Y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.dist_mem_candidates
private theorem dist_mem_candidates :
    (fun p : (X ⊕ Y) × (X ⊕ Y) => dist p.1 p.2) ∈ candidates X Y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: (fun p => dist p.1 p.2) ∈ GromovHausdorff.candidates X Y
-- GromovHausdorff.candidatesBDist_mem_candidatesB
theorem candidatesBDist_mem_candidatesB : candidatesBDist X Y ∈ candidatesB X Y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: GromovHausdorff.candidatesBDist X Y ∈ GromovHausdorff.candidatesB X Y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidatesB_nonempty
private theorem candidatesB_nonempty : (candidatesB X Y).Nonempty 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: (GromovHausdorff.candidatesB X Y).Nonempty
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.closed_candidatesB
private theorem closed_candidatesB : IsClosed (candidatesB X Y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y]
Goal: IsClosed (GromovHausdorff.candidatesB X Y)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.isCompact_candidatesB
private theorem isCompact_candidatesB : IsCompact (candidatesB X Y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [MetricSpace Y] [CompactSpace Y]
Goal: IsCompact (GromovHausdorff.candidatesB X Y)
-- GromovHausdorff.HD_below_aux1
theorem HD_below_aux1 {f : Cb X Y} (C : ℝ) {x : X} :
    BddBelow (range fun y : Y => f (inl x, inr y) + C) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.Cb X Y} (C : ℝ) {x : X}
Goal: BddBelow (Set.range fun y => f (Sum.inl x, Sum.inr y) + C)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_bound_aux1
private theorem HD_bound_aux1 (f : Cb X Y) (C : ℝ) :
    BddAbove (range fun x : X => ⨅ y, f (inl x, inr y) + C) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] [Nonempty Y] (f : GromovHausdorff.Cb X Y) (C : ℝ)
Goal: BddAbove (Set.range fun x => ⨅ y, f (Sum.inl x, Sum.inr y) + C)
-- GromovHausdorff.HD_below_aux2
theorem HD_below_aux2 {f : Cb X Y} (C : ℝ) {y : Y} :
    BddBelow (range fun x : X => f (inl x, inr y) + C) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {f : GromovHausdorff.Cb X Y} (C : ℝ) {y : Y}
Goal: BddBelow (Set.range fun x => f (Sum.inl x, Sum.inr y) + C)
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_bound_aux2
private theorem HD_bound_aux2 (f : Cb X Y) (C : ℝ) :
    BddAbove (range fun y : Y => ⨅ x, f (inl x, inr y) + C) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [Nonempty X] [MetricSpace Y] (f : GromovHausdorff.Cb X Y) (C : ℝ)
Goal: BddAbove (Set.range fun y => ⨅ x, f (Sum.inl x, Sum.inr y) + C)
-- GromovHausdorff.HD_candidatesBDist_le
theorem HD_candidatesBDist_le :
    HD (candidatesBDist X Y) ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: GromovHausdorff.HD (GromovHausdorff.candidatesBDist X Y) ≤ Metric.diam Set.univ + 1 + Metric.diam Set.univ
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_lipschitz_aux1
private theorem HD_lipschitz_aux1 (f g : Cb X Y) :
    (⨆ x, ⨅ y, f (inl x, inr y)) ≤ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [Nonempty X] [MetricSpace Y] [Nonempty Y] (f g : GromovHausdorff.Cb X Y)
Goal: ⨆ x, ⨅ y, f (Sum.inl x, Sum.inr y) ≤ (⨆ x, ⨅ y, g (Sum.inl x, Sum.inr y)) + dist f g
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_lipschitz_aux2
private theorem HD_lipschitz_aux2 (f g : Cb X Y) :
    (⨆ y, ⨅ x, f (inl x, inr y)) ≤ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [Nonempty X] [MetricSpace Y] [Nonempty Y] (f g : GromovHausdorff.Cb X Y)
Goal: ⨆ y, ⨅ x, f (Sum.inl x, Sum.inr y) ≤ (⨆ y, ⨅ x, g (Sum.inl x, Sum.inr y)) + dist f g
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_lipschitz_aux3
private theorem HD_lipschitz_aux3 (f g : Cb X Y) : HD f ≤ HD g + dist f g 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [Nonempty X] [MetricSpace Y] [Nonempty Y] (f g : GromovHausdorff.Cb X Y)
Goal: GromovHausdorff.HD f ≤ GromovHausdorff.HD g + dist f g
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_continuous
private theorem HD_continuous : Continuous (HD : Cb X Y → ℝ) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [Nonempty X] [MetricSpace Y] [Nonempty Y]
Goal: Continuous GromovHausdorff.HD
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.exists_minimizer
private theorem exists_minimizer : ∃ f ∈ candidatesB X Y, ∀ g ∈ candidatesB X Y, HD f ≤ HD g 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: ∃ f ∈ GromovHausdorff.candidatesB X Y,
  ∀ g ∈ GromovHausdorff.candidatesB X Y, GromovHausdorff.HD f ≤ GromovHausdorff.HD g
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.optimalGHDist_mem_candidatesB
private theorem optimalGHDist_mem_candidatesB : optimalGHDist X Y ∈ candidatesB X Y 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: GromovHausdorff.optimalGHDist X Y ∈ GromovHausdorff.candidatesB X Y
-- _private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.HD_optimalGHDist_le
private theorem HD_optimalGHDist_le (g : Cb X Y) (hg : g ∈ candidatesB X Y) :
    HD (optimalGHDist X Y) ≤ HD g 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y] (g : GromovHausdorff.Cb X Y) (hg : g ∈ GromovHausdorff.candidatesB X Y)
Goal: GromovHausdorff.HD (GromovHausdorff.optimalGHDist X Y) ≤ GromovHausdorff.HD g
-- GromovHausdorff.isometry_optimalGHInjl
theorem isometry_optimalGHInjl : Isometry (optimalGHInjl X Y) 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: Isometry (GromovHausdorff.optimalGHInjl X Y)
-- GromovHausdorff.isometry_optimalGHInjr
theorem isometry_optimalGHInjr : Isometry (optimalGHInjr X Y) 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y]
Goal: Isometry (GromovHausdorff.optimalGHInjr X Y)
-- GromovHausdorff.hausdorffDist_optimal_le_HD
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f 
Arguments: (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X] [MetricSpace Y] [CompactSpace Y] [Nonempty Y] {f : GromovHausdorff.Cb X Y} (h : f ∈ GromovHausdorff.candidatesB X Y)
Goal: Metric.hausdorffDist (Set.range (GromovHausdorff.optimalGHInjl X Y)) (Set.range (GromovHausdorff.optimalGHInjr X Y)) ≤
  GromovHausdorff.HD f
-- locallyFinite_of_finite
theorem locallyFinite_of_finite [Finite ι] (f : ι → Set X) : LocallyFinite f 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] [Finite ι] (f : ι → Set X)
Goal: LocallyFinite f
-- LocallyFinite.point_finite
theorem point_finite (hf : LocallyFinite f) (x : X) : { b | x ∈ f b }.Finite 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (x : X)
Goal: {b | x ∈ f b}.Finite
-- LocallyFinite.subset
protected theorem subset (hf : LocallyFinite f) (hg : ∀ i, g i ⊆ f i) : LocallyFinite g 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f g : ι → Set X} (hf : LocallyFinite f) (hg : ∀ (i : ι), g i ⊆ f i)
Goal: LocallyFinite g
-- LocallyFinite.comp_injOn
theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g { i | (f (g i)).Nonempty }) :
    LocallyFinite (f ∘ g) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {g : ι' → ι} (hf : LocallyFinite f) (hg : Set.InjOn g {i | (f (g i)).Nonempty})
Goal: LocallyFinite (f ∘ g)
-- LocallyFinite.comp_injective
theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g) :
    LocallyFinite (f ∘ g) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {g : ι' → ι} (hf : LocallyFinite f) (hg : Function.Injective g)
Goal: LocallyFinite (f ∘ g)
-- locallyFinite_iff_smallSets
theorem _root_.locallyFinite_iff_smallSets :
    LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X}
Goal: LocallyFinite f ↔ ∀ (x : X), ∀ᶠ (s : Set X) in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite
-- LocallyFinite.eventually_smallSets
protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
    ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (x : X)
Goal: ∀ᶠ (s : Set X) in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite
-- LocallyFinite.exists_mem_basis
theorem exists_mem_basis {ι' : Sort*} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
    {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ i, p i ∧ { j | (f j ∩ s i).Nonempty }.Finite 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {ι' : Sort u_6} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X} {x : X} (hb : (𝓝 x).HasBasis p s)
Goal: ∃ i, p i ∧ {j | (f j ∩ s i).Nonempty}.Finite
-- LocallyFinite.nhdsWithin_iUnion
protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
    𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (a : X)
Goal: 𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a
-- LocallyFinite.continuousOn_iUnion'
theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
    (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
    ContinuousOn g (⋃ i, f i) 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (hc : ∀ (i : ι), ∀ x ∈ closure (f i), ContinuousWithinAt g (f i) x)
Goal: ContinuousOn g (⋃ i, f i)
-- LocallyFinite.continuousOn_iUnion
theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
    (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ (i : ι), IsClosed (f i)) (h_cont : ∀ (i : ι), ContinuousOn g (f i))
Goal: ContinuousOn g (⋃ i, f i)
-- LocallyFinite.continuous'
protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ)
    (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
    Continuous g 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = Set.univ) (hc : ∀ (i : ι), ∀ x ∈ closure (f i), ContinuousWithinAt g (f i) x)
Goal: Continuous g
-- LocallyFinite.continuous
protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ)
    (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) :
    Continuous g 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = Set.univ) (h_cl : ∀ (i : ι), IsClosed (f i)) (h_cont : ∀ (i : ι), ContinuousOn g (f i))
Goal: Continuous g
-- LocallyFinite.closure
protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f)
Goal: LocallyFinite fun i => closure (f i)
-- LocallyFinite.closure_iUnion
theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (h : LocallyFinite f)
Goal: closure (⋃ i, f i) = ⋃ i, closure (f i)
-- LocallyFinite.isClosed_iUnion
theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
    IsClosed (⋃ i, f i) 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hc : ∀ (i : ι), IsClosed (f i))
Goal: IsClosed (⋃ i, f i)
-- LocallyFinite.iInter_compl_mem_nhds
theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
    (⋂ (i) (_ : x ∉ f i), (f i)ᶜ) ∈ 𝓝 x 
Arguments: {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hc : ∀ (i : ι), IsClosed (f i)) (x : X)
Goal: ⋂ i, ⋂ (_ : x ∉ f i), (f i)ᶜ ∈ 𝓝 x
-- LocallyFinite.exists_forall_eventually_eq_prod
theorem exists_forall_eventually_eq_prod {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
    (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
    ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2 
Arguments: {X : Type u_4} [TopologicalSpace X] {π : X → Sort u_6} {f : ℕ → (x : X) → π x} (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x})
Goal: ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in Filter.atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2
-- LocallyFinite.exists_forall_eventually_atTop_eventually_eq'
theorem exists_forall_eventually_atTop_eventually_eq' {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
    (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
    ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ n : ℕ in atTop, ∀ᶠ y : X in 𝓝 x, f n y = F y 
Arguments: {X : Type u_4} [TopologicalSpace X] {π : X → Sort u_6} {f : ℕ → (x : X) → π x} (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x})
Goal: ∃ F, ∀ (x : X), ∀ᶠ (n : ℕ) in Filter.atTop, ∀ᶠ (y : X) in 𝓝 x, f n y = F y
-- LocallyFinite.exists_forall_eventually_atTop_eventuallyEq
theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α}
    (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
    ∃ F : X → α, ∀ x, ∀ᶠ n : ℕ in atTop, f n =ᶠ[𝓝 x] F 
Arguments: {α : Type u_3} {X : Type u_4} [TopologicalSpace X] {f : ℕ → X → α} (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x})
Goal: ∃ F, ∀ (x : X), ∀ᶠ (n : ℕ) in Filter.atTop, f n =ᶠ[𝓝 x] F
-- LocallyFinite.preimage_continuous
theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) :
    LocallyFinite (g ⁻¹' f ·) 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g)
Goal: LocallyFinite fun x => g ⁻¹' f x
-- LocallyFinite.prod_right
theorem prod_right (hf : LocallyFinite f) (g : ι → Set Y) : LocallyFinite (fun i ↦ f i ×ˢ g i) 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} (hf : LocallyFinite f) (g : ι → Set Y)
Goal: LocallyFinite fun i => f i ×ˢ g i
-- LocallyFinite.prod_left
theorem prod_left {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set X) :
    LocallyFinite (fun i ↦ f i ×ˢ g i) 
Arguments: {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set X)
Goal: LocallyFinite fun i => f i ×ˢ g i
-- Equiv.locallyFinite_comp_iff
@[simp]
theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f 
Arguments: {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (e : ι' ≃ ι)
Goal: LocallyFinite (f ∘ ⇑e) ↔ LocallyFinite f
-- locallyFinite_sum
theorem locallyFinite_sum {f : ι ⊕ ι' → Set X} :
    LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι ⊕ ι' → Set X}
Goal: LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr)