ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducible
theorem NoetherianSpace.exists_finite_set_closeds_irreducible [NoetherianSpace α] (s : Closeds α) :
    ∃ S : Set (Closeds α), S.Finite ∧ (∀ t ∈ S, IsIrreducible (t : Set α)) ∧ s = sSup S 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] (s : TopologicalSpace.Closeds α)
Goal: ∃ S, S.Finite ∧ (∀ t ∈ S, IsIrreducible ↑t) ∧ s = sSup S
-- TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible
theorem NoetherianSpace.exists_finite_set_isClosed_irreducible [NoetherianSpace α]
    {s : Set α} (hs : IsClosed s) : ∃ S : Set (Set α), S.Finite ∧
      (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] {s : Set α} (hs : IsClosed s)
Goal: ∃ S, S.Finite ∧ (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S
-- TopologicalSpace.NoetherianSpace.exists_finset_irreducible
theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
    ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] (s : TopologicalSpace.Closeds α)
Goal: ∃ S, (∀ (k : { x // x ∈ S }), IsIrreducible ↑↑k) ∧ s = S.sup id
-- TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
    (irreducibleComponents α).Finite 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α]
Goal: (irreducibleComponents α).Finite
-- TopologicalSpace.NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent
theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
    (Z : Set α) (H : Z ∈ irreducibleComponents α) :
    ∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] (Z : Set α) (H : Z ∈ irreducibleComponents α)
Goal: ∃ o, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z
-- continuousOn_Icc_extendFrom_Ioo
theorem continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
    [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β}
    (hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la))
    (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : ContinuousOn (extendFrom (Ioo a b) f) (Icc a b) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β} (hab : a ≠ b) (hf : ContinuousOn f (Set.Ioo a b)) (ha : Filter.Tendsto f (𝓝[>] a) (𝓝 la)) (hb : Filter.Tendsto f (𝓝[<] b) (𝓝 lb))
Goal: ContinuousOn (extendFrom (Set.Ioo a b) f) (Set.Icc a b)
-- eq_lim_at_left_extendFrom_Ioo
theorem eq_lim_at_left_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
    [OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {la : β} (hab : a < b)
    (ha : Tendsto f (𝓝[>] a) (𝓝 la)) : extendFrom (Ioo a b) f a = la 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {la : β} (hab : a < b) (ha : Filter.Tendsto f (𝓝[>] a) (𝓝 la))
Goal: extendFrom (Set.Ioo a b) f a = la
-- eq_lim_at_right_extendFrom_Ioo
theorem eq_lim_at_right_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
    [OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b)
    (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : extendFrom (Ioo a b) f b = lb 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hb : Filter.Tendsto f (𝓝[<] b) (𝓝 lb))
Goal: extendFrom (Set.Ioo a b) f b = lb
-- continuousOn_Ico_extendFrom_Ioo
theorem continuousOn_Ico_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
    [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β}
    (hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) :
    ContinuousOn (extendFrom (Ioo a b) f) (Ico a b) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β} (hab : a < b) (hf : ContinuousOn f (Set.Ioo a b)) (ha : Filter.Tendsto f (𝓝[>] a) (𝓝 la))
Goal: ContinuousOn (extendFrom (Set.Ioo a b) f) (Set.Ico a b)
-- continuousOn_Ioc_extendFrom_Ioo
theorem continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
    [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β}
    (hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) :
    ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hf : ContinuousOn f (Set.Ioo a b)) (hb : Filter.Tendsto f (𝓝[<] b) (𝓝 lb))
Goal: ContinuousOn (extendFrom (Set.Ioo a b) f) (Set.Ioc a b)
-- DenseInducing.dense
structure DenseInducing [TopologicalSpace α] [TopologicalSpace β] (i : α → β)
    extends Inducing i : Prop where
    protected dense : DenseRange i
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (self : DenseInducing i)
Goal: DenseRange i
-- DenseInducing.nhds_eq_comap
theorem nhds_eq_comap (di : DenseInducing i) : ∀ a : α, 𝓝 a = comap i (𝓝 <| i a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) (a : α)
Goal: 𝓝 a = Filter.comap i (𝓝 (i a))
-- DenseInducing.continuous
protected theorem continuous (di : DenseInducing i) : Continuous i 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i)
Goal: Continuous i
-- DenseInducing.closure_range
theorem closure_range : closure (range i) = univ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i)
Goal: closure (Set.range i) = Set.univ
-- DenseInducing.preconnectedSpace
protected theorem preconnectedSpace [PreconnectedSpace α] (di : DenseInducing i) :
    PreconnectedSpace β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [PreconnectedSpace α] (di : DenseInducing i)
Goal: PreconnectedSpace β
-- DenseInducing.closure_image_mem_nhds
theorem closure_image_mem_nhds {s : Set α} {a : α} (di : DenseInducing i) (hs : s ∈ 𝓝 a) :
    closure (i '' s) ∈ 𝓝 (i a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} {s : Set α} {a : α} (di : DenseInducing i) (hs : s ∈ 𝓝 a)
Goal: closure (i '' s) ∈ 𝓝 (i a)
-- DenseInducing.dense_image
theorem dense_image (di : DenseInducing i) {s : Set α} : Dense (i '' s) ↔ Dense s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) {s : Set α}
Goal: Dense (i '' s) ↔ Dense s
-- DenseInducing.interior_compact_eq_empty
theorem interior_compact_eq_empty [T2Space β] (di : DenseInducing i) (hd : Dense (range i)ᶜ)
    {s : Set α} (hs : IsCompact s) : interior s = ∅ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [T2Space β] (di : DenseInducing i) (hd : Dense (Set.range i)ᶜ) {s : Set α} (hs : IsCompact s)
Goal: interior s = ∅
-- DenseInducing.prod
protected theorem prod [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : α → β} {e₂ : γ → δ}
    (de₁ : DenseInducing e₁) (de₂ : DenseInducing e₂) :
    DenseInducing fun p : α × γ => (e₁ p.1, e₂ p.2) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : α → β} {e₂ : γ → δ} (de₁ : DenseInducing e₁) (de₂ : DenseInducing e₂)
Goal: DenseInducing fun p => (e₁ p.1, e₂ p.2)
-- DenseInducing.separableSpace
protected theorem separableSpace [SeparableSpace α] : SeparableSpace β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) [TopologicalSpace.SeparableSpace α]
Goal: TopologicalSpace.SeparableSpace β
-- DenseInducing.tendsto_comap_nhds_nhds
theorem tendsto_comap_nhds_nhds {d : δ} {a : α} (di : DenseInducing i)
    (H : Tendsto h (𝓝 d) (𝓝 (i a))) (comm : h ∘ g = i ∘ f) : Tendsto f (comap g (𝓝 d)) (𝓝 a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace δ] {f : γ → α} {g : γ → δ} {h : δ → β} {d : δ} {a : α} (di : DenseInducing i) (H : Filter.Tendsto h (𝓝 d) (𝓝 (i a))) (comm : h ∘ g = i ∘ f)
Goal: Filter.Tendsto f (Filter.comap g (𝓝 d)) (𝓝 a)
-- DenseInducing.nhdsWithin_neBot
protected theorem nhdsWithin_neBot (di : DenseInducing i) (b : β) : NeBot (𝓝[range i] b) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) (b : β)
Goal: (𝓝[Set.range i] b).NeBot
-- DenseInducing.comap_nhds_neBot
theorem comap_nhds_neBot (di : DenseInducing i) (b : β) : NeBot (comap i (𝓝 b)) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) (b : β)
Goal: (Filter.comap i (𝓝 b)).NeBot
-- DenseInducing.extend_eq_of_tendsto
theorem extend_eq_of_tendsto [T2Space γ] {b : β} {c : γ} {f : α → γ}
    (hf : Tendsto f (comap i (𝓝 b)) (𝓝 c)) : di.extend f b = c 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) [TopologicalSpace γ] [T2Space γ] {b : β} {c : γ} {f : α → γ} (hf : Filter.Tendsto f (Filter.comap i (𝓝 b)) (𝓝 c))
Goal: di.extend f b = c
-- DenseInducing.extend_eq_at
theorem extend_eq_at [T2Space γ] {f : α → γ} {a : α} (hf : ContinuousAt f a) :
    di.extend f (i a) = f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) [TopologicalSpace γ] [T2Space γ] {f : α → γ} {a : α} (hf : ContinuousAt f a)
Goal: di.extend f (i a) = f a
-- DenseInducing.extend_eq_at'
theorem extend_eq_at' [T2Space γ] {f : α → γ} {a : α} (c : γ) (hf : Tendsto f (𝓝 a) (𝓝 c)) :
    di.extend f (i a) = f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) [TopologicalSpace γ] [T2Space γ] {f : α → γ} {a : α} (c : γ) (hf : Filter.Tendsto f (𝓝 a) (𝓝 c))
Goal: di.extend f (i a) = f a
-- DenseInducing.extend_eq
theorem extend_eq [T2Space γ] {f : α → γ} (hf : Continuous f) (a : α) : di.extend f (i a) = f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : DenseInducing i) [TopologicalSpace γ] [T2Space γ] {f : α → γ} (hf : Continuous f) (a : α)
Goal: di.extend f (i a) = f a
-- DenseInducing.extend_eq'
theorem extend_eq' [T2Space γ] {f : α → γ} (di : DenseInducing i)
    (hf : ∀ b, ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)) (a : α) : di.extend f (i a) = f a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] {f : α → γ} (di : DenseInducing i) (hf : ∀ (b : β), ∃ c, Filter.Tendsto f (Filter.comap i (𝓝 b)) (𝓝 c)) (a : α)
Goal: di.extend f (i a) = f a
-- DenseInducing.extend_unique_at
theorem extend_unique_at [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : DenseInducing i)
    (hf : ∀ᶠ x in comap i (𝓝 b), g (i x) = f x) (hg : ContinuousAt g b) : di.extend f b = g b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : DenseInducing i) (hf : ∀ᶠ (x : α) in Filter.comap i (𝓝 b), g (i x) = f x) (hg : ContinuousAt g b)
Goal: di.extend f b = g b
-- DenseInducing.extend_unique
theorem extend_unique [T2Space γ] {f : α → γ} {g : β → γ} (di : DenseInducing i)
    (hf : ∀ x, g (i x) = f x) (hg : Continuous g) : di.extend f = g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] {f : α → γ} {g : β → γ} (di : DenseInducing i) (hf : ∀ (x : α), g (i x) = f x) (hg : Continuous g)
Goal: di.extend f = g
-- DenseInducing.continuousAt_extend
theorem continuousAt_extend [T3Space γ] {b : β} {f : α → γ} (di : DenseInducing i)
    (hf : ∀ᶠ x in 𝓝 b, ∃ c, Tendsto f (comap i <| 𝓝 x) (𝓝 c)) : ContinuousAt (di.extend f) b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T3Space γ] {b : β} {f : α → γ} (di : DenseInducing i) (hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Filter.Tendsto f (Filter.comap i (𝓝 x)) (𝓝 c))
Goal: ContinuousAt (di.extend f) b
-- DenseInducing.continuous_extend
theorem continuous_extend [T3Space γ] {f : α → γ} (di : DenseInducing i)
    (hf : ∀ b, ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)) : Continuous (di.extend f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T3Space γ] {f : α → γ} (di : DenseInducing i) (hf : ∀ (b : β), ∃ c, Filter.Tendsto f (Filter.comap i (𝓝 b)) (𝓝 c))
Goal: Continuous (di.extend f)
-- DenseInducing.mk'
theorem mk' (i : α → β) (c : Continuous i) (dense : ∀ x, x ∈ closure (range i))
    (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ b, i b ∈ t → b ∈ s) : DenseInducing i 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (i : α → β) (c : Continuous i) (dense : ∀ (x : β), x ∈ closure (Set.range i)) (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ (b : α), i b ∈ t → b ∈ s)
Goal: DenseInducing i
-- DenseEmbedding.inj
structure DenseEmbedding [TopologicalSpace α] [TopologicalSpace β] (e : α → β) extends
  DenseInducing e : Prop where
    inj : Function.Injective e
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (self : DenseEmbedding e)
Goal: Function.Injective e
-- DenseEmbedding.mk'
theorem DenseEmbedding.mk' [TopologicalSpace α] [TopologicalSpace β] (e : α → β) (c : Continuous e)
    (dense : DenseRange e) (inj : Function.Injective e)
    (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (e a), ∀ b, e b ∈ t → b ∈ s) : DenseEmbedding e 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α → β) (c : Continuous e) (dense : DenseRange e) (inj : Function.Injective e) (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (e a), ∀ (b : α), e b ∈ t → b ∈ s)
Goal: DenseEmbedding e
-- DenseEmbedding.inj_iff
theorem inj_iff {x y} : e x = e y ↔ x = y 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (de : DenseEmbedding e) {x y : α}
Goal: e x = e y ↔ x = y
-- DenseEmbedding.to_embedding
theorem to_embedding : Embedding e 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (de : DenseEmbedding e)
Goal: Embedding e
-- DenseEmbedding.separableSpace
protected theorem separableSpace [SeparableSpace α] : SeparableSpace β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (de : DenseEmbedding e) [TopologicalSpace.SeparableSpace α]
Goal: TopologicalSpace.SeparableSpace β
-- DenseEmbedding.prod
protected theorem prod {e₁ : α → β} {e₂ : γ → δ} (de₁ : DenseEmbedding e₁)
    (de₂ : DenseEmbedding e₂) : DenseEmbedding fun p : α × γ => (e₁ p.1, e₂ p.2) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : α → β} {e₂ : γ → δ} (de₁ : DenseEmbedding e₁) (de₂ : DenseEmbedding e₂)
Goal: DenseEmbedding fun p => (e₁ p.1, e₂ p.2)
-- DenseEmbedding.subtypeEmb_coe
@[simps]
def subtypeEmb {α : Type*} (p : α → Prop) (e : α → β) (x : { x // p x }) :
    { x // x ∈ closure (e '' { x | p x }) } 
Arguments: {β : Type u_2} [TopologicalSpace β] {α : Type u_5} (p : α → Prop) (e : α → β) (x : { x // p x })
Goal: ↑(DenseEmbedding.subtypeEmb p e x) = e ↑x
-- DenseEmbedding.subtype
protected theorem subtype (p : α → Prop) : DenseEmbedding (subtypeEmb p e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (de : DenseEmbedding e) (p : α → Prop)
Goal: DenseEmbedding (DenseEmbedding.subtypeEmb p e)
-- DenseEmbedding.dense_image
theorem dense_image {s : Set α} : Dense (e '' s) ↔ Dense s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : α → β} (de : DenseEmbedding e) {s : Set α}
Goal: Dense (e '' s) ↔ Dense s
-- denseEmbedding_id
theorem denseEmbedding_id {α : Type*} [TopologicalSpace α] : DenseEmbedding (id : α → α) 
Arguments: {α : Type u_5} [TopologicalSpace α]
Goal: DenseEmbedding id
-- Dense.denseEmbedding_val
theorem Dense.denseEmbedding_val [TopologicalSpace α] {s : Set α} (hs : Dense s) :
    DenseEmbedding ((↑) : s → α) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : Dense s)
Goal: DenseEmbedding Subtype.val
-- isClosed_property
theorem isClosed_property [TopologicalSpace β] {e : α → β} {p : β → Prop} (he : DenseRange e)
    (hp : IsClosed { x | p x }) (h : ∀ a, p (e a)) : ∀ b, p b 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → Prop} (he : DenseRange e) (hp : IsClosed {x | p x}) (h : ∀ (a : α), p (e a)) (b : β)
Goal: p b
-- isClosed_property2
theorem isClosed_property2 [TopologicalSpace β] {e : α → β} {p : β → β → Prop} (he : DenseRange e)
    (hp : IsClosed { q : β × β | p q.1 q.2 }) (h : ∀ a₁ a₂, p (e a₁) (e a₂)) : ∀ b₁ b₂, p b₁ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2}) (h : ∀ (a₁ a₂ : α), p (e a₁) (e a₂)) (b₁ b₂ : β)
Goal: p b₁ b₂
-- isClosed_property3
theorem isClosed_property3 [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop}
    (he : DenseRange e) (hp : IsClosed { q : β × β × β | p q.1 q.2.1 q.2.2 })
    (h : ∀ a₁ a₂ a₃, p (e a₁) (e a₂) (e a₃)) : ∀ b₁ b₂ b₃, p b₁ b₂ b₃ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2.1 q.2.2}) (h : ∀ (a₁ a₂ a₃ : α), p (e a₁) (e a₂) (e a₃)) (b₁ b₂ b₃ : β)
Goal: p b₁ b₂ b₃
-- DenseRange.induction_on
@[elab_as_elim]
theorem DenseRange.induction_on [TopologicalSpace β] {e : α → β} (he : DenseRange e) {p : β → Prop}
    (b₀ : β) (hp : IsClosed { b | p b }) (ih : ∀ a : α, p <| e a) : p b₀ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} (he : DenseRange e) {p : β → Prop} (b₀ : β) (hp : IsClosed {b | p b}) (ih : ∀ (a : α), p (e a))
Goal: p b₀
-- DenseRange.induction_on₂
@[elab_as_elim]
theorem DenseRange.induction_on₂ [TopologicalSpace β] {e : α → β} {p : β → β → Prop}
    (he : DenseRange e) (hp : IsClosed { q : β × β | p q.1 q.2 }) (h : ∀ a₁ a₂, p (e a₁) (e a₂))
    (b₁ b₂ : β) : p b₁ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2}) (h : ∀ (a₁ a₂ : α), p (e a₁) (e a₂)) (b₁ b₂ : β)
Goal: p b₁ b₂
-- DenseRange.induction_on₃
@[elab_as_elim]
theorem DenseRange.induction_on₃ [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop}
    (he : DenseRange e) (hp : IsClosed { q : β × β × β | p q.1 q.2.1 q.2.2 })
    (h : ∀ a₁ a₂ a₃, p (e a₁) (e a₂) (e a₃)) (b₁ b₂ b₃ : β) : p b₁ b₂ b₃ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop} (he : DenseRange e) (hp : IsClosed {q | p q.1 q.2.1 q.2.2}) (h : ∀ (a₁ a₂ a₃ : α), p (e a₁) (e a₂) (e a₃)) (b₁ b₂ b₃ : β)
Goal: p b₁ b₂ b₃
-- DenseRange.equalizer
theorem DenseRange.equalizer (hfd : DenseRange f) {g h : β → γ} (hg : Continuous g)
    (hh : Continuous h) (H : g ∘ f = h ∘ f) : g = h 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β] [TopologicalSpace γ] [T2Space γ] {f : α → β} (hfd : DenseRange f) {g h : β → γ} (hg : Continuous g) (hh : Continuous h) (H : g ∘ f = h ∘ f)
Goal: g = h
-- Filter.HasBasis.hasBasis_of_denseInducing
theorem Filter.HasBasis.hasBasis_of_denseInducing [TopologicalSpace α] [TopologicalSpace β]
    [T3Space β] {ι : Type*} {s : ι → Set α} {p : ι → Prop} {x : α} (h : (𝓝 x).HasBasis p s)
    {f : α → β} (hf : DenseInducing f) : (𝓝 (f x)).HasBasis p fun i => closure <| f '' s i 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [T3Space β] {ι : Type u_5} {s : ι → Set α} {p : ι → Prop} {x : α} (h : (𝓝 x).HasBasis p s) {f : α → β} (hf : DenseInducing f)
Goal: (𝓝 (f x)).HasBasis p fun i => closure (f '' s i)
-- ContinuousInf.continuous_inf
class ContinuousInf (L : Type*) [TopologicalSpace L] [Inf L] : Prop where
    continuous_inf : Continuous fun p : L × L => p.1 ⊓ p.2
Arguments: {L : Type u_1} [TopologicalSpace L] [Inf L] [self : ContinuousInf L]
Goal: Continuous fun p => p.1 ⊓ p.2
-- ContinuousSup.continuous_sup
class ContinuousSup (L : Type*) [TopologicalSpace L] [Sup L] : Prop where
    continuous_sup : Continuous fun p : L × L => p.1 ⊔ p.2
Arguments: {L : Type u_1} [TopologicalSpace L] [Sup L] [self : ContinuousSup L]
Goal: Continuous fun p => p.1 ⊔ p.2
-- continuous_inf
@[continuity]
theorem continuous_inf [Inf L] [ContinuousInf L] : Continuous fun p : L × L => p.1 ⊓ p.2 
Arguments: {L : Type u_1} [TopologicalSpace L] [Inf L] [ContinuousInf L]
Goal: Continuous fun p => p.1 ⊓ p.2
-- Continuous.inf
@[continuity, fun_prop]
theorem Continuous.inf [Inf L] [ContinuousInf L] {f g : X → L} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x => f x ⊓ g x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Inf L] [ContinuousInf L] {f g : X → L} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x ⊓ g x
-- continuous_sup
@[continuity]
theorem continuous_sup [Sup L] [ContinuousSup L] : Continuous fun p : L × L => p.1 ⊔ p.2 
Arguments: {L : Type u_1} [TopologicalSpace L] [Sup L] [ContinuousSup L]
Goal: Continuous fun p => p.1 ⊔ p.2
-- Continuous.sup
@[continuity, fun_prop]
theorem Continuous.sup [Sup L] [ContinuousSup L] {f g : X → L} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x => f x ⊔ g x 
Arguments: {L : Type u_1} {X : Type u_2} [TopologicalSpace L] [TopologicalSpace X] [Sup L] [ContinuousSup L] {f g : X → L} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x ⊔ g x
-- Filter.Tendsto.sup_nhds'
lemma sup_nhds' [Sup L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (f ⊔ g) l (𝓝 (x ⊔ y)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {α : Type u_3} {l : Filter α} {f g : α → L} {x y : L} [Sup L] [ContinuousSup L] (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (f ⊔ g) l (𝓝 (x ⊔ y))
-- Filter.Tendsto.sup_nhds
lemma sup_nhds [Sup L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (fun i => f i ⊔ g i) l (𝓝 (x ⊔ y)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {α : Type u_3} {l : Filter α} {f g : α → L} {x y : L} [Sup L] [ContinuousSup L] (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (fun i => f i ⊔ g i) l (𝓝 (x ⊔ y))
-- Filter.Tendsto.inf_nhds'
lemma inf_nhds' [Inf L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (f ⊓ g) l (𝓝 (x ⊓ y)) 
Arguments: {L : Type u_1} [TopologicalSpace L] {α : Type u_3} {l : Filter α} {f g : α → L} {x y : L} [Inf L] [ContinuousInf L] (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (f ⊓ g) l (𝓝 (x ⊓ y))