ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousAddMonoidHom.sum_map_toFun
@[to_additive (attr := simps!) "Product of two continuous homomorphisms on different spaces."]
def prod_map (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) :
    ContinuousMonoidHom (A × B) (C × D) 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [AddMonoid A] [AddMonoid B] [AddMonoid C] [AddMonoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousAddMonoidHom A C) (g : ContinuousAddMonoidHom B D) (i : A × B)
Goal: (f.sum_map g) i = (f.toAddMonoidHom i.1, g.toAddMonoidHom i.2)
-- ContinuousMonoidHom.prod_map_toFun
@[to_additive (attr := simps!) "Product of two continuous homomorphisms on different spaces."]
def prod_map (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) :
    ContinuousMonoidHom (A × B) (C × D) 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid A] [Monoid B] [Monoid C] [Monoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) (i : A × B)
Goal: (f.prod_map g) i = (f.toMonoidHom i.1, g.toMonoidHom i.2)
-- ContinuousAddMonoidHom.zero_toFun
@[to_additive (attr := simps!) "The trivial continuous homomorphism."]
def one : ContinuousMonoidHom A B 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: ∀ (x : A), (ContinuousAddMonoidHom.zero A B) x = 0
-- ContinuousMonoidHom.one_toFun
@[to_additive (attr := simps!) "The trivial continuous homomorphism."]
def one : ContinuousMonoidHom A B 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: ∀ (x : A), (ContinuousMonoidHom.one A B) x = 1
-- ContinuousMonoidHom.id_toFun
@[to_additive (attr := simps!) "The identity continuous homomorphism."]
def id : ContinuousMonoidHom A A 
Arguments: (A : Type u_2) [Monoid A] [TopologicalSpace A] (x : A)
Goal: (ContinuousMonoidHom.id A) x = x
-- ContinuousAddMonoidHom.id_toFun
@[to_additive (attr := simps!) "The identity continuous homomorphism."]
def id : ContinuousMonoidHom A A 
Arguments: (A : Type u_2) [AddMonoid A] [TopologicalSpace A] (x : A)
Goal: (ContinuousAddMonoidHom.id A) x = x
-- ContinuousAddMonoidHom.fst_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by projection onto the first factor."]
def fst : ContinuousMonoidHom (A × B) A 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B)
Goal: (ContinuousAddMonoidHom.fst A B) self = self.1
-- ContinuousMonoidHom.fst_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by projection onto the first factor."]
def fst : ContinuousMonoidHom (A × B) A 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B)
Goal: (ContinuousMonoidHom.fst A B) self = self.1
-- ContinuousAddMonoidHom.snd_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by projection onto the second factor."]
def snd : ContinuousMonoidHom (A × B) B 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B)
Goal: (ContinuousAddMonoidHom.snd A B) self = self.2
-- ContinuousMonoidHom.snd_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by projection onto the second factor."]
def snd : ContinuousMonoidHom (A × B) B 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B)
Goal: (ContinuousMonoidHom.snd A B) self = self.2
-- ContinuousMonoidHom.inl_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by inclusion of the first factor."]
def inl : ContinuousMonoidHom A (A × B) 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A)
Goal: (ContinuousMonoidHom.inl A B) i =
  ((ContinuousMonoidHom.id A).toMonoidHom i, (ContinuousMonoidHom.one A B).toMonoidHom i)
-- ContinuousAddMonoidHom.inl_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by inclusion of the first factor."]
def inl : ContinuousMonoidHom A (A × B) 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A)
Goal: (ContinuousAddMonoidHom.inl A B) i =
  ((ContinuousAddMonoidHom.id A).toAddMonoidHom i, (ContinuousAddMonoidHom.zero A B).toAddMonoidHom i)
-- ContinuousMonoidHom.inr_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by inclusion of the second factor."]
def inr : ContinuousMonoidHom B (A × B) 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B)
Goal: (ContinuousMonoidHom.inr A B) i =
  ((ContinuousMonoidHom.one B A).toMonoidHom i, (ContinuousMonoidHom.id B).toMonoidHom i)
-- ContinuousAddMonoidHom.inr_toFun
@[to_additive (attr := simps!)
  "The continuous homomorphism given by inclusion of the second factor."]
def inr : ContinuousMonoidHom B (A × B) 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B)
Goal: (ContinuousAddMonoidHom.inr A B) i =
  ((ContinuousAddMonoidHom.zero B A).toAddMonoidHom i, (ContinuousAddMonoidHom.id B).toAddMonoidHom i)
-- ContinuousMonoidHom.diag_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by the diagonal embedding."]
def diag : ContinuousMonoidHom A (A × A) 
Arguments: (A : Type u_2) [Monoid A] [TopologicalSpace A] (i : A)
Goal: (ContinuousMonoidHom.diag A) i = ((ContinuousMonoidHom.id A).toMonoidHom i, (ContinuousMonoidHom.id A).toMonoidHom i)
-- ContinuousAddMonoidHom.diag_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by the diagonal embedding."]
def diag : ContinuousMonoidHom A (A × A) 
Arguments: (A : Type u_2) [AddMonoid A] [TopologicalSpace A] (i : A)
Goal: (ContinuousAddMonoidHom.diag A) i =
  ((ContinuousAddMonoidHom.id A).toAddMonoidHom i, (ContinuousAddMonoidHom.id A).toAddMonoidHom i)
-- ContinuousAddMonoidHom.swap_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by swapping components."]
def swap : ContinuousMonoidHom (A × B) (B × A) 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B)
Goal: (ContinuousAddMonoidHom.swap A B) i =
  ((ContinuousAddMonoidHom.snd A B).toAddMonoidHom i, (ContinuousAddMonoidHom.fst A B).toAddMonoidHom i)
-- ContinuousMonoidHom.swap_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by swapping components."]
def swap : ContinuousMonoidHom (A × B) (B × A) 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B)
Goal: (ContinuousMonoidHom.swap A B) i =
  ((ContinuousMonoidHom.snd A B).toMonoidHom i, (ContinuousMonoidHom.fst A B).toMonoidHom i)
-- ContinuousMonoidHom.mul_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by addition."]
def mul : ContinuousMonoidHom (E × E) E 
Arguments: (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E]
Goal: ∀ (a : E × E), (ContinuousMonoidHom.mul E) a = a.1 * a.2
-- ContinuousAddMonoidHom.add_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by addition."]
def mul : ContinuousMonoidHom (E × E) E 
Arguments: (E : Type u_6) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E]
Goal: ∀ (a : E × E), (ContinuousAddMonoidHom.add E) a = a.1 + a.2
-- ContinuousAddMonoidHom.neg_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by negation."]
def inv : ContinuousMonoidHom E E 
Arguments: (E : Type u_6) [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E]
Goal: ∀ (a : E), (ContinuousAddMonoidHom.neg E) a = -a
-- ContinuousMonoidHom.inv_toFun
@[to_additive (attr := simps!) "The continuous homomorphism given by negation."]
def inv : ContinuousMonoidHom E E 
Arguments: (E : Type u_6) [CommGroup E] [TopologicalSpace E] [TopologicalGroup E]
Goal: ∀ (a : E), (ContinuousMonoidHom.inv E) a = a⁻¹
-- ContinuousAddMonoidHom.coprod_toFun
@[to_additive (attr := simps!) "Coproduct of two continuous homomorphisms to the same space."]
def coprod (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) :
    ContinuousMonoidHom (A × B) E 
Arguments: {A : Type u_2} {B : Type u_3} {E : Type u_6} [AddMonoid A] [AddMonoid B] [AddCommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalAddGroup E] (f : ContinuousAddMonoidHom A E) (g : ContinuousAddMonoidHom B E)
Goal: ∀ (a : A × B), (f.coprod g) a = (ContinuousAddMonoidHom.add E).toAddMonoidHom ((f.sum_map g).toAddMonoidHom a)
-- ContinuousMonoidHom.coprod_toFun
@[to_additive (attr := simps!) "Coproduct of two continuous homomorphisms to the same space."]
def coprod (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) :
    ContinuousMonoidHom (A × B) E 
Arguments: {A : Type u_2} {B : Type u_3} {E : Type u_6} [Monoid A] [Monoid B] [CommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [TopologicalGroup E] (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E)
Goal: ∀ (a : A × B), (f.coprod g) a = (ContinuousMonoidHom.mul E).toMonoidHom ((f.prod_map g).toMonoidHom a)
-- ContinuousMonoidHom.inducing_toContinuousMap
@[to_additive]
theorem inducing_toContinuousMap : Inducing (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: Inducing ContinuousMonoidHom.toContinuousMap
-- ContinuousAddMonoidHom.inducing_toContinuousMap
@[to_additive]
theorem inducing_toContinuousMap : Inducing (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: Inducing ContinuousAddMonoidHom.toContinuousMap
-- ContinuousAddMonoidHom.embedding_toContinuousMap
@[to_additive]
theorem embedding_toContinuousMap :
    Embedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: Embedding ContinuousAddMonoidHom.toContinuousMap
-- ContinuousMonoidHom.embedding_toContinuousMap
@[to_additive]
theorem embedding_toContinuousMap :
    Embedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: Embedding ContinuousMonoidHom.toContinuousMap
-- ContinuousAddMonoidHom.range_toContinuousMap
@[to_additive]
lemma range_toContinuousMap :
    Set.range (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) =
      {f : C(A, B) | f 1 = 1 ∧ ∀ x y, f (x * y) = f x * f y} 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: Set.range ContinuousAddMonoidHom.toContinuousMap = {f | f 0 = 0 ∧ ∀ (x y : A), f (x + y) = f x + f y}
-- ContinuousMonoidHom.range_toContinuousMap
@[to_additive]
lemma range_toContinuousMap :
    Set.range (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) =
      {f : C(A, B) | f 1 = 1 ∧ ∀ x y, f (x * y) = f x * f y} 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B]
Goal: Set.range ContinuousMonoidHom.toContinuousMap = {f | f 1 = 1 ∧ ∀ (x y : A), f (x * y) = f x * f y}
-- ContinuousAddMonoidHom.closedEmbedding_toContinuousMap
@[to_additive]
theorem closedEmbedding_toContinuousMap [ContinuousMul B] [T2Space B] :
    ClosedEmbedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) 
Arguments: (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] [ContinuousAdd B] [T2Space B]
Goal: ClosedEmbedding ContinuousAddMonoidHom.toContinuousMap
-- ContinuousMonoidHom.closedEmbedding_toContinuousMap
@[to_additive]
theorem closedEmbedding_toContinuousMap [ContinuousMul B] [T2Space B] :
    ClosedEmbedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) 
Arguments: (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [ContinuousMul B] [T2Space B]
Goal: ClosedEmbedding ContinuousMonoidHom.toContinuousMap
-- ContinuousAddMonoidHom.continuous_of_continuous_uncurry
@[to_additive]
theorem continuous_of_continuous_uncurry {A : Type*} [TopologicalSpace A]
    (f : A → ContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun x y => f x y)) :
    Continuous f 
Arguments: {B : Type u_3} {C : Type u_4} [AddMonoid B] [AddMonoid C] [TopologicalSpace B] [TopologicalSpace C] {A : Type u_7} [TopologicalSpace A] (f : A → ContinuousAddMonoidHom B C) (h : Continuous (Function.uncurry fun x y => (f x) y))
Goal: Continuous f
-- ContinuousMonoidHom.continuous_of_continuous_uncurry
@[to_additive]
theorem continuous_of_continuous_uncurry {A : Type*} [TopologicalSpace A]
    (f : A → ContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun x y => f x y)) :
    Continuous f 
Arguments: {B : Type u_3} {C : Type u_4} [Monoid B] [Monoid C] [TopologicalSpace B] [TopologicalSpace C] {A : Type u_7} [TopologicalSpace A] (f : A → ContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun x y => (f x) y))
Goal: Continuous f
-- ContinuousMonoidHom.continuous_comp
@[to_additive]
theorem continuous_comp [LocallyCompactSpace B] :
    Continuous fun f : ContinuousMonoidHom A B × ContinuousMonoidHom B C => f.2.comp f.1 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [LocallyCompactSpace B]
Goal: Continuous fun f => f.2.comp f.1
-- ContinuousAddMonoidHom.continuous_comp
@[to_additive]
theorem continuous_comp [LocallyCompactSpace B] :
    Continuous fun f : ContinuousMonoidHom A B × ContinuousMonoidHom B C => f.2.comp f.1 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [LocallyCompactSpace B]
Goal: Continuous fun f => f.2.comp f.1
-- ContinuousMonoidHom.continuous_comp_left
@[to_additive]
theorem continuous_comp_left (f : ContinuousMonoidHom A B) :
    Continuous fun g : ContinuousMonoidHom B C => g.comp f 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom A B)
Goal: Continuous fun g => g.comp f
-- ContinuousAddMonoidHom.continuous_comp_left
@[to_additive]
theorem continuous_comp_left (f : ContinuousMonoidHom A B) :
    Continuous fun g : ContinuousMonoidHom B C => g.comp f 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom A B)
Goal: Continuous fun g => g.comp f
-- ContinuousAddMonoidHom.continuous_comp_right
@[to_additive]
theorem continuous_comp_right (f : ContinuousMonoidHom B C) :
    Continuous fun g : ContinuousMonoidHom A B => f.comp g 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} [AddMonoid A] [AddMonoid B] [AddMonoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousAddMonoidHom B C)
Goal: Continuous fun g => f.comp g
-- ContinuousMonoidHom.continuous_comp_right
@[to_additive]
theorem continuous_comp_right (f : ContinuousMonoidHom B C) :
    Continuous fun g : ContinuousMonoidHom A B => f.comp g 
Arguments: {A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : ContinuousMonoidHom B C)
Goal: Continuous fun g => f.comp g
-- ContinuousMonoidHom.locallyCompactSpace_of_equicontinuousAt
@[to_additive]
theorem locallyCompactSpace_of_equicontinuousAt (U : Set X) (V : Set Y)
    (hU : IsCompact U) (hV : V ∈ nhds (1 : Y))
    (h : EquicontinuousAt (fun f : {f : X →* Y | Set.MapsTo f U V} ↦ (f : X → Y)) 1) :
    LocallyCompactSpace (ContinuousMonoidHom X Y) 
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [Group X] [TopologicalGroup X] [UniformSpace Y] [CommGroup Y] [UniformGroup Y] [T0Space Y] [CompactSpace Y] (U : Set X) (V : Set Y) (hU : IsCompact U) (hV : V ∈ nhds 1) (h : EquicontinuousAt (fun f => ⇑↑f) 1)
Goal: LocallyCompactSpace (ContinuousMonoidHom X Y)
-- ContinuousAddMonoidHom.locallyCompactSpace_of_equicontinuousAt
@[to_additive]
theorem locallyCompactSpace_of_equicontinuousAt (U : Set X) (V : Set Y)
    (hU : IsCompact U) (hV : V ∈ nhds (1 : Y))
    (h : EquicontinuousAt (fun f : {f : X →* Y | Set.MapsTo f U V} ↦ (f : X → Y)) 1) :
    LocallyCompactSpace (ContinuousMonoidHom X Y) 
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [AddGroup X] [TopologicalAddGroup X] [UniformSpace Y] [AddCommGroup Y] [UniformAddGroup Y] [T0Space Y] [CompactSpace Y] (U : Set X) (V : Set Y) (hU : IsCompact U) (hV : V ∈ nhds 0) (h : EquicontinuousAt (fun f => ⇑↑f) 0)
Goal: LocallyCompactSpace (ContinuousAddMonoidHom X Y)
-- ContinuousMonoidHom.locallyCompactSpace_of_hasBasis
@[to_additive]
theorem locallyCompactSpace_of_hasBasis (V : ℕ → Set Y)
    (hV : ∀ {n x}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1))
    (hVo : Filter.HasBasis (nhds 1) (fun _ ↦ True) V) :
    LocallyCompactSpace (ContinuousMonoidHom X Y) 
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [Group X] [TopologicalGroup X] [UniformSpace Y] [CommGroup Y] [UniformGroup Y] [T0Space Y] [CompactSpace Y] [LocallyCompactSpace X] (V : ℕ → Set Y) (hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)) (hVo : (nhds 1).HasBasis (fun x => True) V)
Goal: LocallyCompactSpace (ContinuousMonoidHom X Y)
-- ContinuousAddMonoidHom.locallyCompactSpace_of_hasBasis
@[to_additive]
theorem locallyCompactSpace_of_hasBasis (V : ℕ → Set Y)
    (hV : ∀ {n x}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1))
    (hVo : Filter.HasBasis (nhds 1) (fun _ ↦ True) V) :
    LocallyCompactSpace (ContinuousMonoidHom X Y) 
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [AddGroup X] [TopologicalAddGroup X] [UniformSpace Y] [AddCommGroup Y] [UniformAddGroup Y] [T0Space Y] [CompactSpace Y] [LocallyCompactSpace X] (V : ℕ → Set Y) (hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x + x ∈ V n → x ∈ V (n + 1)) (hVo : (nhds 0).HasBasis (fun x => True) V)
Goal: LocallyCompactSpace (ContinuousAddMonoidHom X Y)
-- subset_coborder
lemma subset_coborder :
    s ⊆ coborder s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: s ⊆ coborder s
-- coborder_inter_closure
lemma coborder_inter_closure :
    coborder s ∩ closure s = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: coborder s ∩ closure s = s
-- closure_inter_coborder
lemma closure_inter_coborder :
    closure s ∩ coborder s = s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: closure s ∩ coborder s = s
-- coborder_eq_union_frontier_compl
lemma coborder_eq_union_frontier_compl :
    coborder s = s ∪ (frontier s)ᶜ 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: coborder s = s ∪ (frontier s)ᶜ
-- coborder_eq_univ_iff
lemma coborder_eq_univ_iff :
    coborder s = univ ↔ IsClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: coborder s = Set.univ ↔ IsClosed s
-- coborder_eq_compl_frontier_iff
lemma coborder_eq_compl_frontier_iff :
    coborder s = (frontier s)ᶜ ↔ IsOpen s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: coborder s = (frontier s)ᶜ ↔ IsOpen s
-- IsOpenMap.coborder_preimage_subset
lemma IsOpenMap.coborder_preimage_subset (hf : IsOpenMap f) (s : Set Y) :
    coborder (f ⁻¹' s) ⊆ f ⁻¹' (coborder s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsOpenMap f) (s : Set Y)
Goal: coborder (f ⁻¹' s) ⊆ f ⁻¹' coborder s
-- Continuous.preimage_coborder_subset
lemma Continuous.preimage_coborder_subset (hf : Continuous f) (s : Set Y) :
    f ⁻¹' (coborder s) ⊆ coborder (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) (s : Set Y)
Goal: f ⁻¹' coborder s ⊆ coborder (f ⁻¹' s)
-- coborder_preimage
lemma coborder_preimage (hf : IsOpenMap f) (hf' : Continuous f) (s : Set Y) :
    coborder (f ⁻¹' s) = f ⁻¹' (coborder s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsOpenMap f) (hf' : Continuous f) (s : Set Y)
Goal: coborder (f ⁻¹' s) = f ⁻¹' coborder s
-- OpenEmbedding.coborder_preimage
protected
lemma OpenEmbedding.coborder_preimage (hf : OpenEmbedding f) (s : Set Y) :
    coborder (f ⁻¹' s) = f ⁻¹' (coborder s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : OpenEmbedding f) (s : Set Y)
Goal: coborder (f ⁻¹' s) = f ⁻¹' coborder s
-- isClosed_preimage_val_coborder
lemma isClosed_preimage_val_coborder :
    IsClosed (coborder s ↓∩ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsClosed (coborder s ↓∩ s)
-- IsOpen.isLocallyClosed
lemma IsOpen.isLocallyClosed (hs : IsOpen s) : IsLocallyClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsOpen s)
Goal: IsLocallyClosed s
-- IsClosed.isLocallyClosed
lemma IsClosed.isLocallyClosed (hs : IsClosed s) : IsLocallyClosed s 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsClosed s)
Goal: IsLocallyClosed s
-- IsLocallyClosed.inter
lemma IsLocallyClosed.inter (hs : IsLocallyClosed s) (ht : IsLocallyClosed t) :
    IsLocallyClosed (s ∩ t) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} (hs : IsLocallyClosed s) (ht : IsLocallyClosed t)
Goal: IsLocallyClosed (s ∩ t)
-- IsLocallyClosed.preimage
lemma IsLocallyClosed.preimage {s : Set Y} (hs : IsLocallyClosed s)
    {f : X → Y} (hf : Continuous f) :
    IsLocallyClosed (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (hs : IsLocallyClosed s) {f : X → Y} (hf : Continuous f)
Goal: IsLocallyClosed (f ⁻¹' s)
-- Inducing.isLocallyClosed_iff
nonrec
lemma Inducing.isLocallyClosed_iff {s : Set X}
    {f : X → Y} (hf : Inducing f) :
    IsLocallyClosed s ↔ ∃ s' : Set Y, IsLocallyClosed s' ∧ f ⁻¹' s' = s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Inducing f)
Goal: IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ f ⁻¹' s' = s
-- Embedding.isLocallyClosed_iff
lemma Embedding.isLocallyClosed_iff {s : Set X}
    {f : X → Y} (hf : Embedding f) :
    IsLocallyClosed s ↔ ∃ s' : Set Y, IsLocallyClosed s' ∧ s' ∩ range f = f '' s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} (hf : Embedding f)
Goal: IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ s' ∩ Set.range f = f '' s
-- IsLocallyClosed.image
lemma IsLocallyClosed.image {s : Set X} (hs : IsLocallyClosed s)
    {f : X → Y} (hf : Inducing f) (hf' : IsLocallyClosed (range f)) :
    IsLocallyClosed (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (hs : IsLocallyClosed s) {f : X → Y} (hf : Inducing f) (hf' : IsLocallyClosed (Set.range f))
Goal: IsLocallyClosed (f '' s)
-- isLocallyClosed_tfae
lemma isLocallyClosed_tfae (s : Set X) :
    List.TFAE
    [ IsLocallyClosed s,
      IsOpen (coborder s),
      ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed (U ↓∩ s),
      ∀ x ∈ s, ∃ U, x ∈ U ∧ IsOpen U ∧ U ∩ closure s ⊆ s,
      IsOpen (closure s ↓∩ s)] 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set X)
Goal: [IsLocallyClosed s, IsOpen (coborder s), ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed (U ↓∩ s),
    ∀ x ∈ s, ∃ U, x ∈ U ∧ IsOpen U ∧ U ∩ closure s ⊆ s, IsOpen (closure s ↓∩ s)].TFAE
-- isLocallyClosed_iff_isOpen_coborder
lemma isLocallyClosed_iff_isOpen_coborder : IsLocallyClosed s ↔ IsOpen (coborder s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X}
Goal: IsLocallyClosed s ↔ IsOpen (coborder s)