ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousAffineMap.cont
structure ContinuousAffineMap (R : Type*) {V W : Type*} (P Q : Type*) [Ring R] [AddCommGroup V]
  [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W]
  [TopologicalSpace Q] [AddTorsor W Q] extends P →ᵃ[R] Q where
  cont : Continuous toFun
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (self : P →ᴬ[R] Q)
Goal: Continuous self.toFun
-- ContinuousAffineMap.to_affineMap_injective
theorem to_affineMap_injective {f g : P →ᴬ[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) :
    f = g 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {f g : P →ᴬ[R] Q} (h : f.toAffineMap = g.toAffineMap)
Goal: f = g
-- ContinuousAffineMap.toFun_eq_coe
theorem toFun_eq_coe (f : P →ᴬ[R] Q) : f.toFun = ⇑f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᴬ[R] Q)
Goal: f.toFun = ⇑f
-- ContinuousAffineMap.coe_injective
theorem coe_injective : @Function.Injective (P →ᴬ[R] Q) (P → Q) (⇑) 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q]
Goal: Function.Injective DFunLike.coe
-- ContinuousAffineMap.ext
@[ext]
theorem ext {f g : P →ᴬ[R] Q} (h : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {f g : P →ᴬ[R] Q} (h : ∀ (x : P), f x = g x)
Goal: f = g
-- ContinuousAffineMap.ext_iff
theorem ext_iff {f g : P →ᴬ[R] Q} : f = g ↔ ∀ x, f x = g x 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {f g : P →ᴬ[R] Q}
Goal: f = g ↔ ∀ (x : P), f x = g x
-- ContinuousAffineMap.congr_fun
theorem congr_fun {f g : P →ᴬ[R] Q} (h : f = g) (x : P) : f x = g x 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {f g : P →ᴬ[R] Q} (h : f = g) (x : P)
Goal: f x = g x
-- ContinuousAffineMap.toContinuousMap_coe
@[simp]
theorem toContinuousMap_coe (f : P →ᴬ[R] Q) : f.toContinuousMap = ↑f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᴬ[R] Q)
Goal: f.toContinuousMap = ↑f
-- ContinuousAffineMap.coe_to_affineMap
@[simp] -- Porting note: removed `norm_cast`
theorem coe_to_affineMap (f : P →ᴬ[R] Q) : ((f : P →ᵃ[R] Q) : P → Q) = f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᴬ[R] Q)
Goal: ⇑f.toAffineMap = ⇑f
-- ContinuousAffineMap.coe_to_continuousMap
theorem coe_to_continuousMap (f : P →ᴬ[R] Q) : ((f : C(P, Q)) : P → Q) = f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᴬ[R] Q)
Goal: ⇑↑f = ⇑f
-- ContinuousAffineMap.to_continuousMap_injective
theorem to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
    f = g 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {f g : P →ᴬ[R] Q} (h : ↑f = ↑g)
Goal: f = g
-- ContinuousAffineMap.coe_affineMap_mk
theorem coe_affineMap_mk (f : P →ᵃ[R] Q) (h) : ((⟨f, h⟩ : P →ᴬ[R] Q) : P →ᵃ[R] Q) = f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᵃ[R] Q) (h : Continuous f.toFun)
Goal: { toAffineMap := f, cont := h }.toAffineMap = f
-- ContinuousAffineMap.coe_continuousMap_mk
@[norm_cast]
theorem coe_continuousMap_mk (f : P →ᵃ[R] Q) (h) : ((⟨f, h⟩ : P →ᴬ[R] Q) : C(P, Q)) = ⟨f, h⟩ 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᵃ[R] Q) (h : Continuous f.toFun)
Goal: ↑{ toAffineMap := f, cont := h } = { toFun := ⇑f, continuous_toFun := h }
-- ContinuousAffineMap.coe_mk
@[simp]
theorem coe_mk (f : P →ᵃ[R] Q) (h) : ((⟨f, h⟩ : P →ᴬ[R] Q) : P → Q) = f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᵃ[R] Q) (h : Continuous f.toFun)
Goal: ⇑{ toAffineMap := f, cont := h } = ⇑f
-- ContinuousAffineMap.mk_coe
@[simp]
theorem mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᴬ[R] Q) (h : Continuous f.toFun)
Goal: { toAffineMap := f.toAffineMap, cont := h } = f
-- ContinuousAffineMap.continuous
@[continuity]
protected theorem continuous (f : P →ᴬ[R] Q) : Continuous f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (f : P →ᴬ[R] Q)
Goal: Continuous ⇑f
-- ContinuousAffineMap.coe_const
@[simp]
theorem coe_const (q : Q) : (const R P q : P → Q) = Function.const P q 
Arguments: (R : Type u_1) {V : Type u_2} {W : Type u_3} (P : Type u_4) {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] (q : Q)
Goal: ⇑(ContinuousAffineMap.const R P q) = Function.const P q
-- ContinuousAffineMap.coe_comp
@[simp, norm_cast]
theorem coe_comp (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q) :
    (f.comp g : P → Q₂) = (f : Q → Q₂) ∘ (g : P → Q) 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {W₂ : Type u_6} {Q₂ : Type u_7} [AddCommGroup W₂] [Module R W₂] [TopologicalSpace Q₂] [AddTorsor W₂ Q₂] (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- ContinuousAffineMap.comp_apply
theorem comp_apply (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q) (x : P) : f.comp g x = f (g x) 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {W₂ : Type u_6} {Q₂ : Type u_7} [AddCommGroup W₂] [Module R W₂] [TopologicalSpace Q₂] [AddTorsor W₂ Q₂] (f : Q →ᴬ[R] Q₂) (g : P →ᴬ[R] Q) (x : P)
Goal: (f.comp g) x = f (g x)
-- ContinuousAffineMap.coe_zero
@[norm_cast, simp]
theorem coe_zero : ((0 : P →ᴬ[R] W) : P → W) = 0 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W]
Goal: ⇑0 = 0
-- ContinuousAffineMap.zero_apply
theorem zero_apply (x : P) : (0 : P →ᴬ[R] W) x = 0 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] (x : P)
Goal: 0 x = 0
-- ContinuousAffineMap.coe_smul
@[norm_cast, simp]
theorem coe_smul (t : S) (f : P →ᴬ[R] W) : ⇑(t • f) = t • ⇑f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] {S : Type u_8} [TopologicalSpace W] [Monoid S] [DistribMulAction S W] [SMulCommClass R S W] [ContinuousConstSMul S W] (t : S) (f : P →ᴬ[R] W)
Goal: ⇑(t • f) = t • ⇑f
-- ContinuousAffineMap.smul_apply
theorem smul_apply (t : S) (f : P →ᴬ[R] W) (x : P) : (t • f) x = t • f x 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] {S : Type u_8} [TopologicalSpace W] [Monoid S] [DistribMulAction S W] [SMulCommClass R S W] [ContinuousConstSMul S W] (t : S) (f : P →ᴬ[R] W) (x : P)
Goal: (t • f) x = t • f x
-- ContinuousAffineMap.coe_add
@[norm_cast, simp]
theorem coe_add (f g : P →ᴬ[R] W) : ⇑(f + g) = f + g 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [TopologicalAddGroup W] (f g : P →ᴬ[R] W)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- ContinuousAffineMap.add_apply
theorem add_apply (f g : P →ᴬ[R] W) (x : P) : (f + g) x = f x + g x 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [TopologicalAddGroup W] (f g : P →ᴬ[R] W) (x : P)
Goal: (f + g) x = f x + g x
-- ContinuousAffineMap.coe_sub
@[norm_cast, simp]
theorem coe_sub (f g : P →ᴬ[R] W) : ⇑(f - g) = f - g 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [TopologicalAddGroup W] (f g : P →ᴬ[R] W)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- ContinuousAffineMap.sub_apply
theorem sub_apply (f g : P →ᴬ[R] W) (x : P) : (f - g) x = f x - g x 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [TopologicalAddGroup W] (f g : P →ᴬ[R] W) (x : P)
Goal: (f - g) x = f x - g x
-- ContinuousAffineMap.coe_neg
@[norm_cast, simp]
theorem coe_neg (f : P →ᴬ[R] W) : ⇑(-f) = -f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [TopologicalAddGroup W] (f : P →ᴬ[R] W)
Goal: ⇑(-f) = -⇑f
-- ContinuousAffineMap.neg_apply
theorem neg_apply (f : P →ᴬ[R] W) (x : P) : (-f) x = -f x 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [TopologicalAddGroup W] (f : P →ᴬ[R] W) (x : P)
Goal: (-f) x = -f x
-- ContinuousLinearMap.coe_toContinuousAffineMap
@[simp]
theorem coe_toContinuousAffineMap (f : V →L[R] W) : ⇑f.toContinuousAffineMap = f 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] (f : V →L[R] W)
Goal: ⇑f.toContinuousAffineMap = ⇑f
-- ContinuousLinearMap.toContinuousAffineMap_map_zero
@[simp]
theorem toContinuousAffineMap_map_zero (f : V →L[R] W) : f.toContinuousAffineMap 0 = 0 
Arguments: {R : Type u_1} {V : Type u_2} {W : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] (f : V →L[R] W)
Goal: f.toContinuousAffineMap 0 = 0
-- UniformSpace.Completion.coe_zero
@[norm_cast]
theorem UniformSpace.Completion.coe_zero [Zero α] : ((0 : α) : Completion α) = 0 
Arguments: {α : Type u_3} [UniformSpace α] [Zero α]
Goal: ↑α 0 = 0
-- UniformSpace.Completion.coe_neg
@[norm_cast]
theorem coe_neg (a : α) : ((-a : α) : Completion α) = -a 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (a : α)
Goal: ↑α (-a) = -↑α a
-- UniformSpace.Completion.coe_sub
@[norm_cast]
theorem coe_sub (a b : α) : ((a - b : α) : Completion α) = a - b 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (a b : α)
Goal: ↑α (a - b) = ↑α a - ↑α b
-- UniformSpace.Completion.coe_add
@[norm_cast]
theorem coe_add (a b : α) : ((a + b : α) : Completion α) = a + b 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [UniformAddGroup α] (a b : α)
Goal: ↑α (a + b) = ↑α a + ↑α b
-- UniformSpace.Completion.toCompl_apply
@[simps]
def toCompl : α →+ Completion α 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: ∀ (a : α), UniformSpace.Completion.toCompl a = ↑α a
-- UniformSpace.Completion.continuous_toCompl
theorem continuous_toCompl : Continuous (toCompl : α → Completion α) 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: Continuous ⇑UniformSpace.Completion.toCompl
-- UniformSpace.Completion.denseInducing_toCompl
theorem denseInducing_toCompl : DenseInducing (toCompl : α → Completion α) 
Arguments: (α : Type u_3) [UniformSpace α] [AddGroup α] [UniformAddGroup α]
Goal: DenseInducing ⇑UniformSpace.Completion.toCompl
-- AddMonoidHom.extension_coe
theorem AddMonoidHom.extension_coe [CompleteSpace β] [T0Space β] (f : α →+ β)
    (hf : Continuous f) (a : α) : f.extension hf a = f a 
Arguments: {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] [AddGroup β] [UniformAddGroup β] [CompleteSpace β] [T0Space β] (f : α →+ β) (hf : Continuous ⇑f) (a : α)
Goal: (f.extension hf) (↑α a) = f a
-- AddMonoidHom.continuous_extension
@[continuity]
theorem AddMonoidHom.continuous_extension [CompleteSpace β] [T0Space β] (f : α →+ β)
    (hf : Continuous f) : Continuous (f.extension hf) 
Arguments: {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] [AddGroup β] [UniformAddGroup β] [CompleteSpace β] [T0Space β] (f : α →+ β) (hf : Continuous ⇑f)
Goal: Continuous ⇑(f.extension hf)
-- AddMonoidHom.continuous_completion
@[continuity]
theorem AddMonoidHom.continuous_completion (f : α →+ β) (hf : Continuous f) :
    Continuous (AddMonoidHom.completion f hf : Completion α → Completion β) 
Arguments: {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] [AddGroup β] [UniformAddGroup β] (f : α →+ β) (hf : Continuous ⇑f)
Goal: Continuous ⇑(f.completion hf)
-- AddMonoidHom.completion_coe
theorem AddMonoidHom.completion_coe (f : α →+ β) (hf : Continuous f) (a : α) :
    AddMonoidHom.completion f hf a = f a 
Arguments: {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] [AddGroup β] [UniformAddGroup β] (f : α →+ β) (hf : Continuous ⇑f) (a : α)
Goal: (f.completion hf) (↑α a) = ↑β (f a)
-- AddMonoidHom.completion_zero
theorem AddMonoidHom.completion_zero :
    AddMonoidHom.completion (0 : α →+ β) continuous_const = 0 
Arguments: {α : Type u_3} {β : Type u_4} [UniformSpace α] [AddGroup α] [UniformAddGroup α] [UniformSpace β] [AddGroup β] [UniformAddGroup β]
Goal: AddMonoidHom.completion 0 ⋯ = 0
-- AddMonoidHom.completion_add
theorem AddMonoidHom.completion_add {γ : Type*} [AddCommGroup γ] [UniformSpace γ]
    [UniformAddGroup γ] (f g : α →+ γ) (hf : Continuous f) (hg : Continuous g) :
    AddMonoidHom.completion (f + g) (hf.add hg) =
    AddMonoidHom.completion f hf + AddMonoidHom.completion g hg 
Arguments: {α : Type u_3} [UniformSpace α] [AddGroup α] [UniformAddGroup α] {γ : Type u_5} [AddCommGroup γ] [UniformSpace γ] [UniformAddGroup γ] (f g : α →+ γ) (hf : Continuous ⇑f) (hg : Continuous ⇑g)
Goal: (f + g).completion ⋯ = f.completion hf + g.completion hg
-- dense_iInter_of_isOpen_nat
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
    (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {f : ℕ → Set X} (ho : ∀ (n : ℕ), IsOpen (f n)) (hd : ∀ (n : ℕ), Dense (f n))
Goal: Dense (⋂ n, f n)
-- dense_sInter_of_isOpen
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
    (hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable) (hd : ∀ s ∈ S, Dense s)
Goal: Dense (⋂₀ S)
-- dense_biInter_of_isOpen
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
    (hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) 
Arguments: {X : Type u_1} {α : Type u_2} [TopologicalSpace X] [BaireSpace X] {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s)) (hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s))
Goal: Dense (⋂ s ∈ S, f s)
-- dense_iInter_of_isOpen
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
    (hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) 
Arguments: {X : Type u_1} {ι : Sort u_3} [TopologicalSpace X] [BaireSpace X] [Countable ι] {f : ι → Set X} (ho : ∀ (i : ι), IsOpen (f i)) (hd : ∀ (i : ι), Dense (f i))
Goal: Dense (⋂ s, f s)
-- mem_residual
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {s : Set X}
Goal: s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t
-- eventually_residual
theorem eventually_residual {p : X → Prop} :
    (∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {p : X → Prop}
Goal: (∀ᶠ (x : X) in residual X, p x) ↔ ∃ t, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x
-- dense_of_mem_residual
theorem dense_of_mem_residual {s : Set X} (hs : s ∈ residual X) : Dense s 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {s : Set X} (hs : s ∈ residual X)
Goal: Dense s
-- dense_sInter_of_Gδ
theorem dense_sInter_of_Gδ {S : Set (Set X)} (ho : ∀ s ∈ S, IsGδ s) (hS : S.Countable)
    (hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {S : Set (Set X)} (ho : ∀ s ∈ S, IsGδ s) (hS : S.Countable) (hd : ∀ s ∈ S, Dense s)
Goal: Dense (⋂₀ S)
-- dense_iInter_of_Gδ
theorem dense_iInter_of_Gδ [Countable ι] {f : ι → Set X} (ho : ∀ s, IsGδ (f s))
    (hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) 
Arguments: {X : Type u_1} {ι : Sort u_3} [TopologicalSpace X] [BaireSpace X] [Countable ι] {f : ι → Set X} (ho : ∀ (s : ι), IsGδ (f s)) (hd : ∀ (s : ι), Dense (f s))
Goal: Dense (⋂ s, f s)
-- dense_biInter_of_Gδ
theorem dense_biInter_of_Gδ {S : Set α} {f : ∀ x ∈ S, Set X} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
    (hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ S, f s ‹_›) 
Arguments: {X : Type u_1} {α : Type u_2} [TopologicalSpace X] [BaireSpace X] {S : Set α} {f : (x : α) → x ∈ S → Set X} (ho : ∀ (s : α) (H : s ∈ S), IsGδ (f s H)) (hS : S.Countable) (hd : ∀ (s : α) (H : s ∈ S), Dense (f s H))
Goal: Dense (⋂ s, ⋂ (h : s ∈ S), f s h)
-- Dense.inter_of_Gδ
theorem Dense.inter_of_Gδ {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
    (htc : Dense t) : Dense (s ∩ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s) (htc : Dense t)
Goal: Dense (s ∩ t)
-- IsGδ.dense_iUnion_interior_of_closed
theorem IsGδ.dense_iUnion_interior_of_closed [Countable ι] {s : Set X} (hs : IsGδ s) (hd : Dense s)
    {f : ι → Set X} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
    Dense (⋃ i, interior (f i)) 
Arguments: {X : Type u_1} {ι : Sort u_3} [TopologicalSpace X] [BaireSpace X] [Countable ι] {s : Set X} (hs : IsGδ s) (hd : Dense s) {f : ι → Set X} (hc : ∀ (i : ι), IsClosed (f i)) (hU : s ⊆ ⋃ i, f i)
Goal: Dense (⋃ i, interior (f i))
-- IsGδ.dense_biUnion_interior_of_closed
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set α} {s : Set X} (hs : IsGδ s) (hd : Dense s)
    (ht : t.Countable) {f : α → Set X} (hc : ∀ i ∈ t, IsClosed (f i)) (hU : s ⊆ ⋃ i ∈ t, f i) :
    Dense (⋃ i ∈ t, interior (f i)) 
Arguments: {X : Type u_1} {α : Type u_2} [TopologicalSpace X] [BaireSpace X] {t : Set α} {s : Set X} (hs : IsGδ s) (hd : Dense s) (ht : t.Countable) {f : α → Set X} (hc : ∀ i ∈ t, IsClosed (f i)) (hU : s ⊆ ⋃ i ∈ t, f i)
Goal: Dense (⋃ i ∈ t, interior (f i))
-- IsGδ.dense_sUnion_interior_of_closed
theorem IsGδ.dense_sUnion_interior_of_closed {T : Set (Set X)} {s : Set X} (hs : IsGδ s)
    (hd : Dense s) (hc : T.Countable) (hc' : ∀ t ∈ T, IsClosed t) (hU : s ⊆ ⋃₀ T) :
    Dense (⋃ t ∈ T, interior t) 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {T : Set (Set X)} {s : Set X} (hs : IsGδ s) (hd : Dense s) (hc : T.Countable) (hc' : ∀ t ∈ T, IsClosed t) (hU : s ⊆ ⋃₀ T)
Goal: Dense (⋃ t ∈ T, interior t)
-- dense_biUnion_interior_of_closed
theorem dense_biUnion_interior_of_closed {S : Set α} {f : α → Set X} (hc : ∀ s ∈ S, IsClosed (f s))
    (hS : S.Countable) (hU : ⋃ s ∈ S, f s = univ) : Dense (⋃ s ∈ S, interior (f s)) 
Arguments: {X : Type u_1} {α : Type u_2} [TopologicalSpace X] [BaireSpace X] {S : Set α} {f : α → Set X} (hc : ∀ s ∈ S, IsClosed (f s)) (hS : S.Countable) (hU : ⋃ s ∈ S, f s = Set.univ)
Goal: Dense (⋃ s ∈ S, interior (f s))
-- dense_sUnion_interior_of_closed
theorem dense_sUnion_interior_of_closed {S : Set (Set X)} (hc : ∀ s ∈ S, IsClosed s)
    (hS : S.Countable) (hU : ⋃₀ S = univ) : Dense (⋃ s ∈ S, interior s) 
Arguments: {X : Type u_1} [TopologicalSpace X] [BaireSpace X] {S : Set (Set X)} (hc : ∀ s ∈ S, IsClosed s) (hS : S.Countable) (hU : ⋃₀ S = Set.univ)
Goal: Dense (⋃ s ∈ S, interior s)
-- dense_iUnion_interior_of_closed
theorem dense_iUnion_interior_of_closed [Countable ι] {f : ι → Set X} (hc : ∀ i, IsClosed (f i))
    (hU : ⋃ i, f i = univ) : Dense (⋃ i, interior (f i)) 
Arguments: {X : Type u_1} {ι : Sort u_3} [TopologicalSpace X] [BaireSpace X] [Countable ι] {f : ι → Set X} (hc : ∀ (i : ι), IsClosed (f i)) (hU : ⋃ i, f i = Set.univ)
Goal: Dense (⋃ i, interior (f i))
-- nonempty_interior_of_iUnion_of_closed
theorem nonempty_interior_of_iUnion_of_closed [Nonempty X] [Countable ι] {f : ι → Set X}
    (hc : ∀ i, IsClosed (f i)) (hU : ⋃ i, f i = univ) : ∃ i, (interior <| f i).Nonempty 
Arguments: {X : Type u_1} {ι : Sort u_3} [TopologicalSpace X] [BaireSpace X] [Nonempty X] [Countable ι] {f : ι → Set X} (hc : ∀ (i : ι), IsClosed (f i)) (hU : ⋃ i, f i = Set.univ)
Goal: ∃ i, (interior (f i)).Nonempty
-- LocallyConnectedSpace.open_connected_basis
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
    open_connected_basis : ∀ x, (𝓝 x).HasBasis (fun s : Set α => IsOpen s ∧ x ∈ s ∧ IsConnected s) id
Arguments: {α : Type u_3} [TopologicalSpace α] [self : LocallyConnectedSpace α] (x : α)
Goal: (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id
-- locallyConnectedSpace_iff_open_connected_basis
theorem locallyConnectedSpace_iff_open_connected_basis :
    LocallyConnectedSpace α ↔
      ∀ x, (𝓝 x).HasBasis (fun s : Set α => IsOpen s ∧ x ∈ s ∧ IsConnected s) id 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: LocallyConnectedSpace α ↔ ∀ (x : α), (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id